An arithmetic progression or arithmetic sequence is **a sequence of numbers such that the difference between the consecutive terms is constant**. For instance, the sequence 5, 7, 9, 11, 13, 15,… is an arithmetic progression with a common difference of 2.

Contents

- 1 What are the 5 examples of arithmetic sequence?
- 2 What is arithmetic sequence given example?
- 3 How do you find the arithmetic sequence?
- 4 Is 12345 a arithmetic sequence?
- 5 What is the example of arithmetic?
- 6 What is example of sequence?
- 7 What is the 4 types of sequence?
- 8 What is arithmetic sequence and series?
- 9 Which is not arithmetic sequence?
- 10 What kind of sequence is 1234?
- 11 What is an in arithmetic?
- 12 Arithmetic Sequences and Sums
- 13 Arithmetic Sequence
- 14 Advanced Topic: Summing an Arithmetic Series
- 15 Footnote: Why Does the Formula Work?
- 16 Introduction to Arithmetic Progressions
- 17 Finite Sequence: Definition & Examples – Video & Lesson Transcript
- 18 Examples
- 19 Finding Patterns
- 20 Finding Sums of Finite Arithmetic Series – Sequences and Series (Algebra 2)
- 21 Arithmetic Sequences and Series
- 22 What is an Arithmetic Sequence?
- 23 How an Arithmetic Sequence Works
- 24 Arithmetic Mean
- 25 Other Types of Sequences
- 26 Arithmetic Sequences
- 27 Arithmetic Sequence – Formula, Meaning, Examples
- 28 What is an Arithmetic Sequence?
- 29 Nth Term of Arithmetic Sequence Formula
- 30 Sum of Arithmetic sequence Formula
- 31 Arithmetic Sequence Formulas
- 32 Difference Between Arithmetic and Geometric Sequence
- 33 Solved Examples on Arithmetic Sequence
- 34 FAQs on Arithmetic sequence
- 34.1 What are Arithmetic Sequence Formulas?
- 34.2 How to Find An Arithmetic Sequence?
- 34.3 What is the n thterm of an Arithmetic Sequence Formula?
- 34.4 What is the Sum of an Arithmetic Sequence Formula?
- 34.5 What is the Formula to Find the Common Difference of Arithmetic sequence?
- 34.6 How to Find n in Arithmetic sequence?
- 34.7 How To Find the First Term in Arithmetic sequence?
- 34.8 What is the Difference Between Arithmetic Sequence and Arithmetic Series?
- 34.9 What are the Types of Sequences?
- 34.10 What are the Applications of Arithmetic Sequence?
- 34.11 How to Find the n thTerm in Arithmetic Sequence?
- 34.12 How to Find the Sum of n Terms of Arithmetic Sequence?

- 35 Arithmetic Sequences and Series
- 36 Arithmetic Series
- 37 Arithmetic Sequence: Formula & Definition – Video & Lesson Transcript
- 38 Finding the Terms
- 39 Finding then th Term
- 40 Arithmetic Sequences: Definition & Finding the Common Difference – Video & Lesson Transcript
- 41 Finding the Common Difference
- 42 The Formula
- 43 Arithmetic Sequences – Precalculus

## What are the 5 examples of arithmetic sequence?

= 3, 6, 9, 12,15,. A few more examples of an arithmetic sequence are: 5, 8, 11, 14, 80, 75, 70, 65, 60,

## What is arithmetic sequence given example?

What is an arithmetic sequence? An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term. For example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6. An arithmetic sequence can be known as an arithmetic progression.

## How do you find the arithmetic sequence?

An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence then subtract it by the previous one, and the result is always the same or constant then it is an arithmetic sequence.

## Is 12345 a arithmetic sequence?

For example, the sequence 1, 2, 3, 4, is an arithmetic progression with common difference 1. The distance between any two successive members is called common difference.

## What is the example of arithmetic?

For example, the sequence 3, 5, 7, 9 is arithmetic because the difference between consecutive terms is always two. The sequence 21, 16, 11, 6 is arithmetic as well because the difference between consecutive terms is always minus five.

## What is example of sequence?

more A list of numbers or objects in a special order. Example: 3, 5, 7, 9, is a sequence starting at 3 and increasing by 2 each time.

## What is the 4 types of sequence?

Types of Sequence and Series

- Arithmetic Sequences.
- Geometric Sequences.
- Harmonic Sequences.
- Fibonacci Numbers.

## What is arithmetic sequence and series?

An arithmetic sequence is a sequence where the difference d between successive terms is constant. The general term of an arithmetic sequence can be written in terms of its first term a1, common difference d, and index n as follows: an=a1+(n−1)d. An arithmetic series is the sum of the terms of an arithmetic sequence.

## Which is not arithmetic sequence?

The following are not examples of arithmetic sequences: 1.) 2,4,8,16 is not because the difference between first and second term is 2, but the difference between second and third term is 4, and the difference between third and fourth term is 8. No common difference so it is not an arithmetic sequence.

## What kind of sequence is 1234?

This is an arithmetic sequence since there is a common difference between each term. In this case, adding 1 to the previous term in the sequence gives the next term.

## What is an in arithmetic?

an = a + (n – 1) d. The number d is called the common difference because any two consecutive terms of an. arithmetic sequence differ by d, and it is found by subtracting any pair of terms an and.

## Arithmetic Sequences and Sums

A sequence is a collection of items (typically numbers) that are arranged in a specific order. Each number in the sequence is referred to as aterm (or “element” or “member” in certain cases); for additional information, see Sequences and Series.

## Arithmetic Sequence

An Arithmetic Sequence is characterized by the fact that the difference between one term and the next is a constant. In other words, we just increase the value by the same amount each time. endlessly.

### Example:

1, 4, 7, 10, 13, 16, 19, 22, and 25 are the numbers 1 through 25. Each number in this series has a three-digit gap between them. Each time the pattern is repeated, the last number is increased by three, as seen below: As a general rule, we could write an arithmetic series along the lines of

- There are two words: Ais the first term, and dis is the difference between the two terms (sometimes known as the “common difference”).

### Example: (continued)

1, 4, 7, 10, 13, 16, 19, 22, and 25 are the numbers 1 through 25. Has:

- In this equation, A = 1 represents the first term, while d = 3 represents the “common difference” between terms.

And this is what we get:

### Rule

It is possible to define an Arithmetic Sequence as a rule:x n= a + d(n1) (We use “n1” since it is not used in the first term of the sequence).

### Example: Write a rule, and calculate the 9th term, for this Arithmetic Sequence:

3, 8, 13, 18, 23, 28, 33, and 38 are the numbers three, eight, thirteen, and eighteen. Each number in this sequence has a five-point gap between them. The values ofaanddare as follows:

- A = 3 (the first term)
- D = 5 (the “common difference”)
- A = 3 (the first term).

Making use of the Arithmetic Sequencerule, we can see that_xn= a + d(n1)= 3 + 5(n1)= 3 + 3 + 5n 5 = 5n 2 xn= a + d(n1) = 3 + 3 + 3 + 5n n= 3 + 3 + 3 As a result, the ninth term is:x 9= 5 9 2= 43 Is that what you’re saying? Take a look for yourself! Arithmetic Sequences (also known as Arithmetic Progressions (A.P.’s)) are a type of arithmetic progression.

## Advanced Topic: Summing an Arithmetic Series

To summarize the terms of this arithmetic sequence:a + (a+d) + (a+2d) + (a+3d) + (a+4d) + (a+5d) + (a+6d) + (a+7d) + (a+8d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + ( make use of the following formula: What exactly is that amusing symbol? It is referred to as The Sigma Notation is a type of notation that is used to represent a sigma function. Additionally, the starting and finishing values are displayed below and above it: “Sum upnwherengoes from 1 to 4,” the text states. 10 is the correct answer.

### Example: Add up the first 10 terms of the arithmetic sequence:

The values ofa,dandnare as follows:

- In this equation, A = 1 represents the first term, d = 3 represents the “common difference” between terms, and n = 10 represents the number of terms to add up.

As a result, the equation becomes:= 5(2+93) = 5(29) = 145 Check it out yourself: why don’t you sum up all of the phrases and see whether it comes out to 145?

## Footnote: Why Does the Formula Work?

Let’s take a look at why the formula works because we’ll be employing an unusual “technique” that’s worth understanding. First, we’ll refer to the entire total as “S”: S = a + (a + d) +. + (a + (n2)d) +(a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n1)d) + After that, rewrite S in the opposite order: S = (a + (n1)d)+ (a + (n2)d)+. +(a + d)+a. +(a + d)+a. +(a + d)+a. Now, term by phrase, add these two together:

S | = | a | + | (a+d) | + | . | + | (a + (n-2)d) | + | (a + (n-1)d) |

S | = | (a + (n-1)d) | + | (a + (n-2)d) | + | . | + | (a + d) | + | a |

2S | = | (2a + (n-1)d) | + | (2a + (n-1)d) | + | . | + | (2a + (n-1)d) | + | (2a + (n-1)d) |

Each and every term is the same!

Furthermore, there are “n” of them. 2S = n (2a + (n1)d) = n (2a + (n1)d) Now, we can simply divide by two to obtain the following result: The function S = (n/2) (2a + (n1)d) is defined as This is the formula we’ve come up with:

## Introduction to Arithmetic Progressions

Generally speaking, a progression is a sequence or series of numbers in which the numbers are organized in a certain order so that the relationship between the succeeding terms of a series or sequence remains constant. It is feasible to find the n thterm of a series by following a set of steps. There are three different types of progressions in mathematics:

- Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP) are all types of progression.

AP, also known as Arithmetic Sequence, is a sequence or series of integers in which the common difference between two subsequent numbers in the series is always the same. As an illustration: Series 1: 1, 3, 5, 7, 9, and 11. Every pair of successive integers in this sequence has a common difference of 2, which is always true. Series 2: numbers 28, 25, 22, 19, 16, 13,. There is no common difference between any two successive numbers in this series; instead, there is a strict -3 between any two consecutive numbers.

### Terminology and Representation

- Common difference, d = a 2– a 1= a 3– a 2=. = a n– a n – 1
- A n= n thterm of Arithmetic Progression
- S n= Sum of first n elements in the series
- A n= n

### General Form of an AP

Given that ais treated as a first term anddis treated as a common difference, the N thterm of the AP may be calculated using the following formula: As a result, using the above-mentioned procedure to compute the n terms of an AP, the general form of the AP is as follows: Example: The 35th term in the sequence 5, 11, 17, 23,. is to be found. Solution: When looking at the given series,a = 5, d = a 2– a 1= 11 – 5 = 6, and n = 35 are the values. As a result, we must use the following equations to figure out the 35th term: n= a + (n – 1)da n= 5 + (35 – 1) x 6a n= 5 + 34 x 6a n= 209 As a result, the number 209 represents the 35th term.

### Sum of n Terms of Arithmetic Progression

The arithmetic progression sum is calculated using the formula S n= (n/2)

### Derivation of the Formula

Allowing ‘l’ to signify the n thterm of the series and S n to represent the sun of the first n terms of the series a, (a+d), (a+2d),., a+(n-1)d S n = a 1 plus a 2 plus a 3 plus .a n-1 plus a n S n= a + (a + d) + (a + 2d) +. + (l – 2d) + (l – d) + l. + (l – 2d) + (l – d) + l. + (l – 2d) + (l – d) + l. (1) When we write the series in reverse order, we obtain S n= l + (l – d) + (l – 2d) +. + (a + 2d) + (a + d) + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a … (2) Adding equations (1) and (2) results in equation (2).

+ (a + l) + (a + l) + (a + l) +.

(3) As a result, the formula for calculating the sum of a series is S n= (n/2)(a + l), where an is the first term of the series, l is the last term of the series, and n is the number of terms in the series.

d S n= (n/2)(a + a + (n – 1)d)(a + a + (n – 1)d) S n= (n/2)(2a + (n – 1) x d)(n/2)(2a + (n – 1) x d) Observation: The successive terms in an Arithmetic Progression can alternatively be written as a-3d, a-2d, a-d, a, a+d, a+2d, a+3d, and so on.

### Sample Problems on Arithmetic Progressions

Problem 1: Calculate the sum of the first 35 terms in the sequence 5,11,17,23, and so on. a = 5 in the given series, d = a 2–a in the provided series, and so on. The number 1 equals 11 – 5 = 6, and the number n equals 35. S n= (n/2)(2a + (n – 1) x d)(n/2)(2a + (n – 1) x d) S n= (35/2)(2 x 5 + (35 – 1) x 6)(35/2)(2 x 5 + (35 – 1) x 6) S n= (35/2)(10 + 34 x 6) n= (35/2)(10 + 34 x 6) n= (35/2)(10 + 34 x 6) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) A = 35214/2A = 3745S n= 35214/2A = 3745 Find the sum of a series where the first term of the series is 5 and the last term of the series is 209, and the number of terms in the series is 35, as shown in Problem 2.

Problem 2.

S n= (35/2)(5 + 209) S n= (35/2)(5 + 209) S n= (35/2)(5 + 209) A = 35214/2A = 3745S n= 35214/2A = 3745 Problem 3: A amount of 21 rupees is divided among three brothers, with each of the three pieces of money being in the AP and the sum of their squares being the sum of their squares being 155.

Solution: Assume that the three components of money are (a-d), a, and (a+d), and that the total amount allocated is in AP.

155 divided by two equals 155 Taking the value of ‘a’ into consideration, we obtain 3(7) 2+ 2d.

2= 4d = 2 = 2 The three portions of the money that was dispersed are as follows:a + d = 7 + 2 = 9a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5 As a result, the most significant portion is Rupees 9 million.

## Finite Sequence: Definition & Examples – Video & Lesson Transcript

When there is a finite series, the first term is followed by a second term, and so on until the last term. In a finite sequence, the letternoften reflects the total number of phrases in the sequence. In a finite series, the first term may be represented by a (1), the second term can be represented by a (2), etc. Parentheses are often used to separate numbers adjacent to thea, but parentheses will be used at other points in this course to distinguish them from subscripts. This terminology is illustrated in the following graphic.

## Examples

Despite the fact that there are many other forms of finite sequences, we shall confine ourselves to the field of mathematics for the time being. The prime numbers smaller than 40, for example, are an example of a finite sequence, as seen in the table below: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, Another example is the range of natural numbers between zero and one hundred. Due to the fact that it would be tedious to write down all of the terms in this finite sequence, we will demonstrate it as follows: 1, 2, 3, 4, 5,., 100 are the numbers from 1 to 100.

Alternatively, there might be alternative sequences that begin in the same way and end in 100, but which do not contain the natural numbers less than or equal to 100.

## Finding Patterns

Let’s take a look at some finite sequences to see if there are any patterns.

#### Example 1

An anarithmetic finite sequence is a sequence in which all pairs of succeeding terms share a common difference and is defined as follows: Figure out which of these arithmetic finite sequences has the most common difference: 2, 7, 12, 17,., 47 are the prime numbers. Because the common difference is 5, the first four terms of the series demonstrate that the common difference is 5. With another way of saying it, we may add 5 to any phrase in the series to get the following term in the sequence.

## Finding Sums of Finite Arithmetic Series – Sequences and Series (Algebra 2)

The sum of all terms for a finitearithmetic sequencegiven bywherea1 is the first term,dis the common difference, andnis the number of terms may be determined using the following formula: wherea1 is the first term,dis the common difference, andnis the number of terms Consider the arithmetic sum as an example of how the total is computed in this manner. If you choose not to add all of the words at once, keep in mind that the first and last terms may be recast as2fives. This method may be used to rewrite the second and second-to-last terms as well.

There are 9fives in all, and the aggregate is 9 x 5 = 9.

expand more Due to the fact that the difference between each term is constant, this sequence from 1 through 1000 is arithmetic.

The first phrase, a1, is one and the last term, is one thousand thousand. The total number of terms is less than 1000. The sum of all positive integers up to and including 1000 is 500 500. }}}!}}}Test}}} arrow back} arrow forwardarrow leftarrow right

## Arithmetic Sequences and Series

An arithmetic sequence is a set of integers in which the difference between the words that follow is always the same as its predecessor.

### Learning Objectives

Make a calculation for the nth term of an arithmetic sequence and then define the characteristics of arithmetic sequences.

### Key Takeaways

- When the common differenced is used, the behavior of the arithmetic sequence is determined. Arithmetic sequences may be either limited or infinite in length.

#### Key Terms

- Arithmetic sequence: An ordered list of numbers in which the difference between the subsequent terms is constant
- Endless: An ordered list of numbers in which the difference between the consecutive terms is infinite
- Infinite, unending, without beginning or end
- Limitless
- Innumerable

For example, an arithmetic progression or arithmetic sequence is a succession of integers in which the difference between the following terms is always the same as the difference between the previous terms. A common difference of 2 may be found in the arithmetic sequence 5, 7, 9, 11, 13, cdots, which is an example of an arithmetic sequence.

- 1: The initial term in the series
- D: The difference between the common differences of consecutive terms
- A 1: a n: Then the nth term in the series.

The behavior of the arithmetic sequence is determined by the common differenced arithmetic sequence. If the common difference,d, is the following:

- Positively, the sequence will continue to develop towards infinity (+infty). If the sequence is negative, it will regress towards negative infinity (-infty)
- If it is positive, it will regress towards positive infinity (-infty).

It should be noted that the first term in the series can be thought of asa 1+0cdot d, the second term can be thought of asa 1+1cdot d, and the third term can be thought of asa 1+2cdot d, and therefore the following equation givesa n:a n In the equation a n= a 1+(n1)cdot D Of course, one may always type down each term until one has the term desired—but if one need the 50th term, this can be time-consuming and inefficient.

## What is an Arithmetic Sequence?

Sequences of numbers are useful in algebra because they allow you to see what occurs when something keeps becoming larger or smaller over time. The common difference, which is the difference between one number and the next number in the sequence, is the defining characteristic of an arithmetic sequence. This difference is a constant value in arithmetic sequences, and it can be either positive or negative in nature. Consequently, an arithmetic sequence continues to grow or shrink by a defined amount each time a new number is added to the list of numbers that make up the sequence is added to it.

#### TL;DR (Too Long; Didn’t Read)

As defined by the Common Difference formula, an arithmetic sequence is a list of integers in which consecutive entries differ by the same amount, called the common difference. Whenever the common difference is positive, the sequence continues to grow by a predetermined amount, and when it is negative, the series begins to shrink. The geometric series, in which terms differ by a common factor, and the Fibonacci sequence, in which each number is the sum of the two numbers before it, are two more typical sequences that might be encountered.

## How an Arithmetic Sequence Works

There are three elements that form an arithmetic series: a starting number, a common difference, and the number of words in the sequence. For example, the first twelve terms of an arithmetic series with a common difference of three and five terms are 12, 15, 18, 21, and 24. A declining series starting with the number 3 has a common difference of 2 and six phrases, and it is an example of a decreasing sequence. This series is composed of the numbers 3, 1, 1, 3, 5, and 7.

There is also the possibility of an unlimited number of terms in arithmetic sequences. Examples of infinite number of terms include, for example, the first sequence above with 12 terms, followed by 15 terms, followed by 18 terms, and so on infinity.

## Arithmetic Mean

A matching series to an arithmetic sequence is a series that sums all of the terms in the sequence. When the terms are put together and the total is divided by the total number of terms, the result is the arithmetic mean or the mean of the sum of the terms. The arithmetic mean may be calculated using the formula text = frac n text. The observation that when the first and last terms of an arithmetic sequence are added, the total is the same as when the second and next to last terms are added, or when the third and third to last terms are added, provides a simple method of computing the mean of an arithmetic series.

The mean of an arithmetic sequence is calculated by dividing the total by the number of terms in the sequence; hence, the mean of an arithmetic sequence is half the sum of the first and final terms.

Instead, by restricting the total to a specific number of items, it is possible to find the mean of a partial sum.

## Other Types of Sequences

Observations from experiments or measurements of natural occurrences are frequently used to create numerical sequences. Such sequences can be made up of random numbers, although they are more typically made up of arithmetic or other ordered lists of numbers than random numbers. Geometric sequences, as opposed to arithmetic sequences, vary in that they share a common component rather than a common difference in their composition. To avoid the repetition of the same number being added or deleted for each new phrase, a number is multiplied or divided for each new term that is added.

Other sequences are governed by whole distinct sets of laws.

The numbers are as follows: 1, 1, 2, 3, 5, 8, and so on.

Arithmetic sequences are straightforward, yet they have a variety of practical applications.

## Arithmetic Sequences

In mathematics, an arithmetic sequence is a succession of integers in which the value of each number grows or decreases by a fixed amount each term. When an arithmetic sequence has n terms, we may construct a formula for each term in the form fn+c, where d is the common difference. Once you’ve determined the common difference, you can calculate the value ofcby substituting 1fornand the first term in the series fora1 into the equation. Example 1: The arithmetic sequence 1,5,9,13,17,21,25 is an arithmetic series with a common difference of four.

- For the thenthterm, we substituten=1,a1=1andd=4inan=dn+cto findc, which is the formula for thenthterm.
- As an example, the arithmetic sequence 12-9-6-3-0-3-6-0 is an arithmetic series with a common difference of three.
- It is important to note that, because the series is decreasing, the common difference is a negative number.) To determine the next3 terms, we just keep subtracting3: 6 3=9 9 3=12 12 3=15 6 3=9 9 3=12 12 3=15 As a result, the next three terms are 9, 12, and 15.
- As a result, the formula for the fifteenth term in this series isan=3n+15.

Exemple No. 3: The number series 2,3,5,8,12,17,23,. is not an arithmetic sequence. Differencea2 is 1, but the following differencea3 is 2, and the differencea4 is 3. There is no way to write a formula in the form of forman=dn+c for this sequence. Geometric sequences are another type of sequence.

## Arithmetic Sequence – Formula, Meaning, Examples

When you have a succession of integers where the differences between every two subsequent numbers are the same, you have an arithmetic sequence. Let us take a moment to review what a sequence is. A sequence is a set of integers that are arranged in a certain manner. An arithmetic sequence is defined as follows: 1, 6, 11, 16,. is an arithmetic sequence because it follows a pattern in which each number is acquired by adding 5 to the phrase before it. There are two arithmetic sequence formulae available.

- The formula for determining the nth term of an arithmetic series. An arithmetic series has n terms, and the sum of the first n terms is determined by the following formula:

Let’s look at the definition of an arithmetic sequence, as well as arithmetic sequence formulae, derivations, and a slew of other examples to get us started.

1. | What is an Arithmetic Sequence? |

2. | Terms Related to Arithmetic Sequence |

3. | Nth Term of Arithmetic Sequence Formula |

4. | Sum of Arithmetic sequence Formula |

5. | Arithmetic Sequence Formulas |

6. | Difference Between Arithmetic and Geometric Sequence |

7. | FAQs on Arithmetic sequence |

## What is an Arithmetic Sequence?

There are two ways in which anarithmetic sequence can be defined. When the differences between every two succeeding words are the same, it is said to be in sequence (or) Every term in an arithmetic series is generated by adding a specified integer (either positive or negative, or zero) to the term before it. Here is an example of an arithmetic sequence.

### Arithmetic Sequences Example

For example, consider the series 3, 6, 9, 12, 15, which is an arithmetic sequence since every term is created by adding a constant number (3) to the term immediately before that one. Here,

- A = 3 for the first term
- D = 6 – 3 for the common difference
- 12 – 9 for the second term
- 15 – 12 for the third term
- A = 3 for the third term

As a result, arithmetic sequences can be expressed as a, a + d, a + 2d, a + 3d, and so forth. Let’s use the previous scenario as an example of how to test this pattern. a, a + d, a + 2d, a + 3d, a + 4d,. = 3, 3 + 3, 3 + 2(3), 3 + 3(3), 3 + 4(3),. = 3, 6, 9, 12, 15,. = 3, 6, 9, 12, 15,. = 3, 6, 9, 12, 15,. = 3, 6, 9, 12, 15,. Here are a few more instances of arithmetic sequences to consider:

- 5, 8, 11, 14,
- 80, 75, 70, 65, 60,
- 2/2, 3/2, 2/2,
- -2, -22, -32, -42,
- 5/8, 11/14,

The terms of an arithmetic sequence are often symbolized by the letters a1, a2, a3, and so on. Arithmetic sequences are discussed in the following way, according to the vocabulary we employ.

### First Term of Arithmetic Sequence

The first term of an arithmetic sequence is, as the name implies, the first integer in the sequence. It is often symbolized by the letters a1 (or) a. For example, the first word in the sequence 5, 8, 11, 14, is the number 5. Specifically, a1 = 6 (or) a = 6.

### Common Difference of Arithmetic Sequence

The addition of a fixed number to each preceding term in an arithmetic series, with one exception (the first term), has previously been demonstrated in prior sections. The “fixed number” in this case is referred to as the “common difference,” and it is symbolized by the letter d. The formula for the common difference isd = a – an1.

## Nth Term of Arithmetic Sequence Formula

In such case, the thterm of an arithmetic series of the form A1, A2, A3,. is given byan = a1 + (n-1) d. This is also referred to as the broad word for the arithmetic sequence in some circles. This comes immediately from the notion that the arithmetic sequence a1, a2, a3,. = a1, a1 + d, a1 + 2d, a1 + 3d,. = a1, a1 + d, a1 + 3d,. = a1, a1 + 3d,. = a1, a1 + 3d,. Several arithmetic sequences are shown in the following table, along with the first term, the common difference, and the subsequent n thterms.

Arithmetic sequence | First Term(a) | Common Difference(d) | n thtermaₙ = a₁ + (n – 1) d |
---|---|---|---|

80, 75, 70, 65, 60,. | 80 | -5 | 80 + (n – 1) (-5)= -5n + 85 |

π/2, π, 3π/2, 2π,. | π/2 | π/2 | π/2 + (n – 1) (π/2)= nπ/2 |

-√2, -2√2, -3√2, -4√2,. | -√2 | -√2 | -√2 + (n – 1) (-√2)= -√2 n |

### Arithmetic Sequence Recursive Formula

It is possible to utilize the following formula for finding the nthterm of an arithmetic series in order to discover any term of that sequence if the values of ‘a1′ and’d’ are known, however this is not recommended. One further method of determining what term is the n thterm is to utilize the ” recursive formula of an arithmetic sequence “. This formula may be used to determine the next term (an) of an arithmetic sequence given both its preceding term (an1) and the value of the variable ‘d’ are known.

Example: If a19 = -72 and d = 7, find the value of a21 in an arithmetic sequence. Solution: a20 = a19 + d = -72 + 7 = -65 is obtained by applying the recursive formula. a21 = a20 + d = -65 + 7 = -58; a21 = a20 + d = -65 + 7 = -58; a21 = a20 + d = -65 + 7 = -58; As a result, the value of a21 is -58.

## Sum of Arithmetic sequence Formula

To obtain the sum of the first n terms of an arithmetic sequence, the sum of the arithmetic sequence formula is employed. Consider the following arithmetic sequence: a1 (or ‘a’) is the first term, and d is the common difference between the first and second terms. Sn is the symbol for the sum of the first n terms in the expression. Then

- The following is true: When the n thterm is unknown, Sn= n/2
- When the n thterm is known, Sn= n/2

Example Ms. Natalie makes $200,000 each year, with an annual pay rise of $25,000 in addition to that. So, how much money does she have at the conclusion of the first five years of her career? Solution In Ms. Natalie’s first year of employment, she earns a sum equal to a = 2,000,000. The annual increase is denoted by the symbol d = 25,000. We need to figure out how much money she will make in the first five years. As a result, n = 5. In the sum sum of arithmetic sequence formula, substituting these numbers results in Sn = n/2 Sn = 5/2(2(200000) + (5 – 1)(25000), which is 5/2 (400000 +100000), which is equal to 5/2 (500000), which is equal to 1250000.

We may modify this formula to be more useful for greater values of the constant ‘n.’

### Sum of Arithmetic Sequence Proof

Consider the following arithmetic sequence: a1 is the first term, and d is the common difference between the two terms. The sum of the first ‘n’ terms of the series is given bySn = a1 + (a1 + d) + (a1 + 2d) +. + an, where Sn = a1 + (a1 + d) + (a1 + 2d) +. + an. (1) Let us write the same total from right to left in the same manner (i.e., from the n thterm to the first term). (an – d) + (an – 2d) +. + a1. Sn = a plus (an – d) plus (an – 2d) +. + a1. (2)By combining (1) and (2), all words beginning with the letter ‘d’ are eliminated.

+ (a1 + an) 2Sn = n (a1 + an) = n (a1 + an) Sn =/2 is a mathematical expression.

## Arithmetic Sequence Formulas

The following are the formulae that are connected to the arithmetic sequence.

- There is a common distinction, the n-th phrase, a = (a + 1)d
- The sum of n terms, Sn =/2 (or) n/2 (2a + 1)d
- The n-th term, a = (a + 1)d
- The n-th term, a = a + (n-1)d

## Difference Between Arithmetic and Geometric Sequence

The following are the distinctions between arithmetic sequence and geometric sequence:

Arithmetic sequences | Geometric sequences |

In this, the differences between every two consecutive numbers are the same. | In this, theratiosof every two consecutive numbers are the same. |

It is identified by the first term (a) and the common difference (d). | It is identified by the first term (a) and the common ratio (r). |

There is a linear relationship between the terms. | There is an exponential relationship between the terms. |

Notes on the Arithmetic Sequence that are very important

- Arithmetic sequences have the same difference between every two subsequent numbers
- This is known as the difference between two consecutive numbers. The common difference of an arithmetic sequence a1, a2, and a3 is d = a2 – a1 = a3 – a2 =
- The common difference of an arithmetic sequence a1, a2, and a3 is d = a2 – a1 = a3 – a2 =
- It is an= a1 + (n1)d for the n-th term of an integer arithmetic sequence. It is equal to n/2 when the sum of the first n terms of an arithmetic sequence is calculated. Positive, negative, or zero can be used to represent the common difference of arithmetic sequences.

Arithmetic Sequence-Related Discussion Topics

- Sequence Calculator, Series Calculator, Arithmetic Sequence Calculator, Geometric Sequence Calculator are all terms used to refer to the same thing.

## Solved Examples on Arithmetic Sequence

- Examples: Find the nth term in the arithmetic sequence -5, -7/2, -2 and the nth term in the arithmetic sequence Solution: The numbers in the above sequence are -5, -7/2, -2, and. There are two terms in this equation: the first is equal to -5, and the common difference is equal to -(7/2) – (-5) = -2 – (-7/2) = 3/2. The n thterm of an arithmetic sequence can be calculated using the formulaan = a + b. (n – 1) dan = -5 +(n – 1) (3/2)= -5+ (3/2)n – 3/2= 3n/2 – 13/2 = dan = -5 +(n – 1) (3/2)= dan = -5 +(n – 1) (3/2)= dan = -5 +(n – 1) (3/2)= dan = -5 +(3/2)n – 3/2= dan = -5 +(3/2)n – 3/2= dan = Example 2:Which term of the arithmetic sequence -3, -8, -13, -18, the answer is: the specified arithmetic sequence is: 3, 8, 13, 18, and so on. The first term is represented by the symbol a = -3. The common difference is d = -8 – (-3) = -13 – (-8) = -5. The common difference is d = -8 – (-3) = -13 – (-8) = -5. It has been established that the n thterm is a = -248. All of these values should be substituted in the n th l term of an arithmetic sequence formula,an = a + b. (n – 1) d-248 equals -3 plus (-5) (n – 1) the sum of -248 and 248 equals 3 -5n, and the sum of 5n and 250 equals -5nn equals 50. Answer: The number 248 represents the 50th phrase in the provided sequence.

Continue to the next slide proceed to the next slide Simple graphics might help you break through difficult topics. Math will no longer be a difficult topic for you, especially if you visualize the concepts and grasp them as a result. Schedule a No-Obligation Trial Class.

## FAQs on Arithmetic sequence

An arithmetic sequence is a sequence of integers in which every term (with the exception of the first term) is generated by adding a constant number to the preceding term. For example, the arithmetic sequence 1, 3, 5, 7, is an arithmetic sequence because each term is created by adding 2 (a constant integer) to the term before it.

### What are Arithmetic Sequence Formulas?

Here are the formulae connected to an arithmetic series where a1 (or a) is the first term and d is the common difference: a1 (or a) is the first term, and d is the common difference:

- When we look at the common difference, it is second term minus first term. The n thterm of the series is defined as a = a + (n – 1)d
- Sn =/2 (or) n/2 (2a + (n – 1)d) is the sum of the n terms in the sequence.

### How to Find An Arithmetic Sequence?

Whenever the difference between every two successive terms of a series is the same, then the sequence is said to be an arithmetic sequence. For example, the numbers 3, 8, 13, and 18 are arithmetic because

### What is the n thterm of an Arithmetic Sequence Formula?

The n thterm of arithmetic sequences is represented by the expression a = a + (n – 1) d. The letter ‘a’ stands for the first term, while the letter ‘d’ stands for the common difference.

### What is the Sum of an Arithmetic Sequence Formula?

Arithmetic sequences with a common difference ‘d’ and the first term ‘a’ are denoted by Sn, and we have two formulae to compute the sum of the first n terms with the common difference ‘d’.

### What is the Formula to Find the Common Difference of Arithmetic sequence?

As the name implies, the common difference of an arithmetic sequence is the difference between every two of its consecutive (or consecutively occurring) terms. Finding the common difference of an arithmetic series may be calculated using the formula: d = a – an1.

### How to Find n in Arithmetic sequence?

When we are asked to find the number of terms (n) in arithmetic sequences, it is possible that part of the information about a, d, an, or Sn has already been provided in the problem. We will simply substitute the supplied values in the formulae of an or Sn and solve for n as a result of this.

### How To Find the First Term in Arithmetic sequence?

The number that appears in the first position from the left of an arithmetic sequence is referred to as the first term of the sequence. It is symbolized by the letter ‘a’. If the letter ‘a’ is not provided in the problem, then the problem may contain some information concerning the letter d (or) the letter a (or) the letter Sn. We shall simply insert the specified values in the formulae of an or Sn and solve for a by dividing by two.

### What is the Difference Between Arithmetic Sequence and Arithmetic Series?

The number that appears in the first position from the left in an arithmetic sequence is referred to as the first term. A ‘a’ is used to signify that it is present. If the letter ‘a’ is not provided in the issue, then the problem may contain some information about d (or) a (or) Sn. In order to solve for ‘a’, we will simply insert the supplied values into the an or Sn formulae.

### What are the Types of Sequences?

In mathematics, there are three basic types of sequences. They are as follows:

- The arithmetic series, the geometric sequence, and the harmonic sequence are all examples of sequences.

### What are the Applications of Arithmetic Sequence?

Here are some examples of applications: The pay of a person who receives an annual raise of a fixed amount, the rent of a taxi that charges by the mile traveled, the number of fish in a pond that increases by a certain number each month, and so on are examples of steady increases.

### How to Find the n thTerm in Arithmetic Sequence?

The following are the actions to take in order to get the n thterm of arithmetic sequences:

- Identify the first term, a
- The common difference, d
- And the last term, e. Choose the word that you wish to use. n, to be precise. All of them should be substituted into the formula a = a + (n – 1) d

### How to Find the Sum of n Terms of Arithmetic Sequence?

The first term is a; the second term is d; the third term is e; and the fourth term is Choose the phrase that you wish to use in your essay. n, to be precise Fill in the blanks with the values from the table a = a + (n – 1) d

- Identify the initial term (a)
- The common difference (d)
- And the last term (e). Determine which phrase you wish to use (n)
- All of them should be substituted into the formula Sn= n/2(2a + (n – 1)d)

## Arithmetic Sequences and Series

The succession of arithmetic operations There is a series of integers in which each subsequent number is equal to the sum of the preceding number and specified constants. orarithmetic progression is a type of progression in which numbers are added together. This term is used to describe a series of integers in which each subsequent number is the sum of the preceding number and a certain number of constants (e.g., 1). an=an−1+d Sequence of Arithmetic Operations Furthermore, becauseanan1=d, the constant is referred to as the common difference.

For example, the series of positive odd integers is an arithmetic sequence, consisting of the numbers 1, 3, 5, 7, 9, and so on.

This word may be constructed using the generic terman=an1+2where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, To formulate the following equation in general terms, given the initial terma1of an arithmetic series and its common differenced, we may write: a2=a1+da3=a2+d=(a1+d)+d=a1+2da4=a3+d=(a1+2d)+d=a1+3da5=a4+d=(a1+3d)+d=a1+4d⋮ As a result, we can see that each arithmetic sequence may be expressed as follows in terms of its initial element, common difference, and index: an=a1+(n−1)d Sequence of Arithmetic Operations In fact, every generic word that is linear defines an arithmetic sequence in its simplest definition.

### Example 1

Identify the general term of the above arithmetic sequence and use that equation to determine the series’s 100th term. For example: 7,10,13,16,19,… Solution: The first step is to determine the common difference, which is d=10 7=3. It is important to note that the difference between any two consecutive phrases is three. The series is, in fact, an arithmetic progression, with a1=7 and d=3. an=a1+(n1)d=7+(n1)3=7+3n3=3n+4 and an=a1+(n1)d=7+(n1)3=7+3n3=3n+4 and an=a1+(n1)d=3 As a result, we may express the general terman=3n+4 as an equation.

To determine the 100th term, use the following equation: a100=3(100)+4=304 Answer_an=3n+4;a100=304 It is possible that the common difference of an arithmetic series be negative.

### Example 2

Identify the general term of the given arithmetic sequence and use it to determine the 75th term of the series: 6,4,2,0,−2,… Solution: Make a start by determining the common difference, d = 4 6=2. Next, determine the formula for the general term, wherea1=6andd=2 are the variables. an=a1+(n−1)d=6+(n−1)⋅(−2)=6−2n+2=8−2n As a result, an=8nand the 75thterm may be determined as follows: an=8nand the 75thterm a75=8−2(75)=8−150=−142 Answer_an=8−2n;a100=−142 The terms in an arithmetic sequence that occur between two provided terms are referred to as arithmetic means.

### Example 3

Locate the general term of the given arithmetic sequence and use it to determine the 75th term of the series: 6,4,2,0,−2,… Solution: Find the common difference, d=4+6+2 to get started. Afterwards, determine the general term’s formula, which isa1=6andd=2. an=a1+(n−1)d=6+(n−1)⋅(−2)=6−2n+2=8−2n The 75 thterm may be computed as follows: an=8n and the 75 thterm are equal to an=8n. a75=8−2(75)=8−150=−142 Answer_an=8−2n;a100=−142 Arithmetic means are the terms that appear between two supplied words in an arithmetic sequence.

### Example 4

Find the general term of an arithmetic series with a3=1 and a10=48 as the first and last terms. Solution: We’ll need a1 and d in order to come up with a formula for the general term. Using the information provided, it is possible to construct a linear system using these variables as variables. andan=a1+(n−1) d:{a3=a1+(3−1)da10=a1+(10−1)d⇒ {−1=a1+2d48=a1+9d Make use of a3=1. Make use of a10=48. Multiplying the first equation by one and adding the result to the second equation will eliminate a1.

an=a1+(n−1)d=−15+(n−1)⋅7=−15+7n−7=−22+7n Answer_an=7n−22 Take a look at this! Identify the general term of the above arithmetic sequence and use that equation to determine the series’s 100th term. For example: 32,2,52,3,72,… Answer_an=12n+1;a100=51

## Arithmetic Series

Series of mathematical operations When an arithmetic sequence is added together, the result is called the sum of its terms (or the sum of its terms and numbers). Consider the following sequence: S5=n=15(2n1)=++++= 1+3+5+7+9+25=25, where S5=n=15(2n1)=++++ = 1+3+5+7+9=25, where S5=n=15(2n1)=++++= 1+3+5+7+9 = 25. Adding 5 positive odd numbers together, like we have done previously, is manageable and straightforward. Consider, on the other hand, adding the first 100 positive odd numbers. This would be quite time-consuming.

When we write this series in reverse, we get Sn=an+(and)+(an2d)+.+a1 as a result.

2.:Sn=n(a1+an) 2 Calculate the sum of the first 100 terms of the sequence defined byan=2n1 by using this formula.

The sum of the two variables, S100, is 100 (1 + 100)2 = 100(1 + 199)2.

### Example 5

The sum of the first 50 terms of the following sequence: 4, 9, 14, 19, 24,. is to be found. The solution is to determine whether or not there is a common difference between the concepts that have been provided. d=9−4=5 It is important to note that the difference between any two consecutive phrases is 5. The series is, in fact, an arithmetic progression, and we may writean=a1+(n1)d=4+(n1)5=4+5n5=5n1 as an anagram of the sequence. As a result, the broad phrase isan=5n1 is used. For this sequence, we need the 1st and 50th terms to compute the 50thpartial sum of the series: a1=4a50=5(50)−1=249 Then, using the formula, find the partial sum of the given arithmetic sequence that is 50th in length.

### Example 6

Evaluate:Σn=135(10−4n). This problem asks us to find the sum of the first 35 terms of an arithmetic series with a general terman=104n. The solution is as follows: This may be used to determine the 1 stand for the 35th period. a1=10−4(1)=6a35=10−4(35)=−130 Then, using the formula, find out what the 35th partial sum will be. Sn=n(a1+an)2S35=35⋅(a1+a35)2=352=35(−124)2=−2,170 2,170 is the answer.

### Example 7

In an outdoor amphitheater, the first row of seating comprises 26 seats, the second row contains 28 seats, the third row contains 30 seats, and so on and so forth. Is there a maximum capacity for seating in the theater if there are 18 rows of seats? The Roman Theater (Fig. 9.2) (Wikipedia) Solution: To begin, discover a formula that may be used to calculate the number of seats in each given row. In this case, the number of seats in each row is organized into a sequence: 26,28,30,… It is important to note that the difference between any two consecutive words is 2.

where a1=26 and d=2.

As a result, the number of seats in each row may be calculated using the formulaan=2n+24.

In order to do this, we require the following 18 thterms: a1=26a18=2(18)+24=60 This may be used to calculate the 18th partial sum, which is calculated as follows: Sn=n(a1+an)2S18=18⋅(a1+a18)2=18(26+60) 2=9(86)=774 There are a total of 774 seats available.

Take a look at this! Calculate the sum of the first 60 terms of the following sequence of numbers: 5, 0, 5, 10, 15,. are all possible combinations. Answer_S60=−8,550

### Key Takeaways

- When the difference between successive terms is constant, a series is called an arithmetic sequence. According to the following formula, the general term of an arithmetic series may be represented as the sum of its initial term, common differenced term, and indexnumber, as follows: an=a1+(n−1)d
- An arithmetic series is the sum of the terms of an arithmetic sequence
- An arithmetic sequence is the sum of the terms of an arithmetic series
- As a result, the partial sum of an arithmetic series may be computed using the first and final terms in the following manner: Sn=n(a1+an)2

### Topic Exercises

- Given the first term and common difference of an arithmetic series, write the first five terms of the sequence. Calculate the general term for the following numbers: a1=5
- D=3
- A1=12
- D=2
- A1=15
- D=5
- A1=7
- D=4
- D=1
- A1=23
- D=13
- A 1=1
- D=12
- A1=54
- D=14
- A1=1.8
- D=0.6
- A1=4.3
- D=2.1

- Find a formula for the general term based on the arithmetic sequence and apply it to get the 100 th term based on the series. 0.8, 2, 3.2, 4.4, 5.6,.
- 4.4, 7.5, 13.7, 16.8,.
- 3, 8, 13, 18, 23,.
- 3, 7, 11, 15, 19,.
- 6, 14, 22, 30, 38,.
- 5, 10, 15, 20, 25,.
- 2, 4, 6, 8, 10,.
- 12,52,92,132,.
- 13, 23, 53,83,.
- 14,12,54,2,114,. Find the positive odd integer that is 50th
- Find the positive even integer that is 50th
- Find the 40 th term in the sequence that consists of every other positive odd integer in the following format: 1, 5, 9, 13,.
- Find the 40th term in the sequence that consists of every other positive even integer: 1, 5, 9, 13,.
- Find the 40th term in the sequence that consists of every other positive even integer: 2, 6, 10, 14,.
- 2, 6, 10, 14,. What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
- What number is the phrase 172 in the arithmetic sequence 4, 4, 12, 20, 28,.
- What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
- Find an equation that yields the general term in terms of a1 and the common differenced given the arithmetic sequence described by the recurrence relationan=an1+5wherea1=2 andn1 and the common differenced
- Find an equation that yields the general term in terms ofa1and the common differenced, given the arithmetic sequence described by the recurrence relationan=an1-9wherea1=4 andn1
- This is the problem.

- Calculate a formula for the general term based on the terms of an arithmetic sequence: a1=6anda7=42
- A1=12anda12=6
- A1=19anda26=56
- A1=9anda31=141
- A1=16anda10=376
- A1=54anda11=654
- A3=6anda26=40
- A3=16andananda15=

- Find all possible arithmetic means between the given terms: a1=3anda6=17
- A1=5anda5=7
- A2=4anda8=7
- A5=12anda9=72
- A5=15anda7=21
- A6=4anda11=1
- A7=4anda11=1

### Part B: Arithmetic Series

- Find all arithmetic means between the given terms: a1=3anda6=17
- A1=5anda5=7
- A2=4anda8=7
- A5=12anda9=72
- A5=15anda7=21
- A6=4anda11=1
- A7=4anda11=1

- Consider the following values: n=1160(3n)
- N=1121(2n)
- N=1250(4n3)
- N=1120(2n+12)
- N=170(198n)
- N=1220(5n)
- N=160(5212n)
- N=151(38n+14)
- N=1120(1.5n2.6)
- N=1175(0.2n1.6)
- The total of all 200 positive integers is found by counting them up. To solve this problem, find the sum of the first 400 positive integers.

- The generic term for a sequence of positive odd integers is denoted byan=2n1 and is defined as follows: Furthermore, the generic phrase for a sequence of positive even integers is denoted by the number an=2n. Look for the following: The sum of the first 50 positive odd integers
- The sum of the first 200 positive odd integers
- The sum of the first 50 positive even integers
- The sum of the first 200 positive even integers
- The sum of the first 100 positive even integers
- The sum of the firstk positive odd integers
- The sum of the firstk positive odd integers the sum of the firstk positive even integers
- The sum of the firstk positive odd integers
- There are eight seats in the front row of a tiny theater, which is the standard configuration. Following that, each row contains three additional seats than the one before it. How many total seats are there in the theater if there are 12 rows of seats? In an outdoor amphitheater, the first row of seating comprises 42 seats, the second row contains 44 seats, the third row contains 46 seats, and so on and so forth. When there are 22 rows, how many people can fit in the theater’s entire seating capacity? The number of bricks in a triangle stack are as follows: 37 bricks on the bottom row, 34 bricks on the second row and so on, ending with one brick on the top row. What is the total number of bricks in the stack
- Each succeeding row of a triangle stack of bricks contains one fewer brick, until there is just one brick remaining on the top of the stack. Given a total of 210 bricks in the stack, how many rows does the stack have? A wage contract with a 10-year term pays $65,000 in the first year, with a $3,200 raise for each consecutive year after. Calculate the entire salary obligation over a ten-year period (see Figure 1). In accordance with the hour, a clock tower knocks its bell a specified number of times. The clock strikes once at one o’clock, twice at two o’clock, and so on until twelve o’clock. A day’s worth of time is represented by the number of times the clock tower’s bell rings.

### Part C: Discussion Board

- Is the Fibonacci sequence an arithmetic series or a geometric sequence? How to explain: Using the formula for the then th partial sum of an arithmetic sequenceSn=n(a1+an)2and the formula for the general terman=a1+(n1)dto derive a new formula for the then th partial sum of an arithmetic sequenceSn=n2, we can derive the formula for the then th partial sum of an arithmetic sequenceSn=n2. How would this formula be beneficial in the given situation? Explain with the use of an example of your own creation
- Discuss strategies for computing sums in situations when the index does not begin with one. For example, n=1535(3n+4)=1,659
- N=1535(3n+4)=1,659
- Carl Friedrich Gauss is the subject of a well-known tale about his misbehaving in school. As a punishment, his instructor assigned him the chore of adding the first 100 integers to his list of disciplinary actions. According to folklore, young Gauss replied accurately within seconds of being asked. The question is, what is the solution, and how do you believe he was able to come up with the figure so quickly?

### Answers

- 5, 8, 11, 14, 17
- An=3n+2
- 15, 10, 5, 0, 0
- An=205n
- 12,32,52,72,92
- An=n12
- 1,12, 0,12, 1
- An=3212n
- 1.8, 2.4, 3, 3.6, 4.2
- An=0.6n+1.2
- An=6n3
- A100=597
- An=14n
- A100=399
- An=5n
- A100=500
- An=2n32

- 5, 8, 11, 14, 17
- An=3n+2
- 15, 10, 5, 0, 0
- An=20n
- 12,32,52,72,92
- An=n12
- 1,12, 0,12, 1
- An=3212n
- 1.8, 2.4, 3, 3.6, 4.2
- An=0.6n+1.2
- An=6n3
- A100=597
- An=14n
- A100=399
- An=5n
- A100=500
- An=2n32
- A

## Arithmetic Sequence: Formula & Definition – Video & Lesson Transcript

8, 11, 14, 17;an=3n+2; 15, 10, 5, 0, 0;an=20n; 12,32,52,72,92;an=n12; 1,12, 0,12, 1;an=3212n; 1.8, 2.4, 3, 3.6, 4.2;an=0.6n+1.2; an=6n3;a100=597; an=14n;a100=399; an=5n;a100=500; an=2n32;a100

## Finding the Terms

Let’s start with a straightforward problem. We have the following numbers in our sequence: -3, 2, 7, 12,. What is the seventh and last phrase in this sequence? As we can see, the most typical difference between successive periods is five points. The fourth term equals twelve, therefore a (4) = twelve. We can continue to add terms to the list in the following order until we reach the seventh term: -3, 2, 7, 12, 17, 22, 27,. and so on. This tells us that a (7) = 27 is the answer.

## Finding then th Term

Consider the identical sequence as in the preceding example, with the exception that we must now discover the 33rd word oracle (33). We may utilize the same strategy as previously, but it would take a long time to complete the project. We need to come up with a way that is both faster and more efficient. We are aware that we are starting with ata (1), which is a negative number. We multiply each phrase by 5 to get the next term. To go from a (1) to a (33), we’d have to add 32 consecutive terms (33 – 1 = 32) to the beginning of the sequence.

To put it another way, a (33) = -3 + (33 – 1)5.

a (33) = -3 + (33 – 1)5 = -3 + 160 = 157.

Then the relationship between the th term and the initial terma (1) and the common differencedis provided by:

## Arithmetic Sequences: Definition & Finding the Common Difference – Video & Lesson Transcript

The candy bar example provides the numbers 2, 4, and 6 as a starting point. If just two people showed up to the party, we may call it a night. However, if we wanted to, we could keep the pattern going indefinitely as well. The process may be continued by multiplying 2 by 6 to obtain 8, and then multiplying 2 by 8 to get 10. We’d get numbers like 2, 4, 6, 8, 10, and so on. There are several other instances of arithmetic sequences, including: 1, 2, 3, 4,. and so on. 2, 5, 8, 11,. and so on.

and so on are the numbers.

## Finding the Common Difference

We can compute the common difference for each of our sequences by choosing any two integers that are adjacent to each other and subtracting the first from the second. This method works for all of our sequences. We may repeat the process with another set of numbers to ensure that the difference remains the same. For our first series of 1, 2, 3, 4,., we can subtract the 1 from the 2 to get 2 – 1 = 1. For our second sequence of 1, 2, 3, 4,., we can subtract the 1 from the 2 to get 2 – 1 = 1.

Take a look at it!

Repeating this process with the 3 and the 4 will reveal that it too has a difference of 1, indicating that this arithmetic sequence has one common difference of one.

When we deduct the 5 from the 8 and the 8 from the 11, we get a total of 3 as well.

In this case, we obtain 2 if we remove 3 from 5 and 5 from 7 respectively. The result of subtracting 7 from 9 is similarly 2. As a result, the most common difference for this sequence is number 2.

## The Formula

Given that we have a common difference between all of the numbers in our arithmetic series, we can utilize this knowledge to develop a formula that will allow us to locate any number in our sequence, whether it is the tenth number or the fifty-first number in our sequence. It’s important to remember that each number in an arithmetic sequence is actually the first number plus the common difference multiplied by the number of times we have to add it up to get there. Consider how we arrived at the second term by first adding the common difference to the first term once:

The common difference must be added once to the first term to get to the second term.## Arithmetic Sequences – Precalculus

Sequences, probability, and counting theory are all topics covered in this course.

### Learning Objectives

You will learn the following things in this section:

- You will learn the following things in this section: a.

Large purchases, such as computers and automobiles, are frequently made by businesses for their own use. For taxation reasons, the book-value of these supplies diminishes with each passing year. Depreciation is the term used to describe this decline in value. Depreciation may be calculated in several ways, one of which is straight-line depreciation, which means that the value of the asset drops by the same amount each year. Consider the case of a lady who decides to start her own modest contracting firm.

She expects to be able to sell the truck for $8,000 after five years, according to her estimations.

After one year, the vehicle will be worth?21,600; after two years, it will be worth?18,200; after three years, it will be worth?14,800; after four years, it will be worth?11,400; and after five years, it will be worth?8,000.

### Finding Common Differences

The values of the vehicle in the example are said to constitute an anarithmetic sequence since they vary by a consistent amount each year, according to the definition. Every term grows or decreases by the same constant amount, which is referred to as the common difference of the sequence. –3,400 is the common difference between the two sequences in this case. Another example of an arithmetic series may be seen in the sequence below. In this situation, the constant difference is three times more than one.

Sequence of Arithmetic Operations When two successive words are added together, the difference between them is a constant.

Given that and is the initial term of an arithmetic sequence, and is the common difference, then the sequence will be as follows: Identifying Commonalities and Dissimilarities Is each of the sequences mathematical in nature?

To establish whether or not there is a common difference between two terms, subtract each phrase from the succeeding term.

- The series is not arithmetic because there is no common difference between the elements
- The sequence is arithmetic because there is a common difference between the elements. The most often encountered difference is 4

Analysis Each of these sequences is represented by a graph, which is depicted in (Figure). We can observe from the graphs that, despite the fact that both sequences exhibit increase, is not linear whereas is linear, as we previously said. Given that arithmetic sequences have an invariant rate of change, their graphs will always consist of points on a straight line. If we are informed that a series is arithmetic, do we have to subtract every term from the term after it in order to identify the common difference between the terms?

- As long as we know that the sequence is arithmetic, we may take any one term from it and subtract it from the following term to determine the common difference.
- If this is the case, identify the common difference.
- The main distinction is whether or not the provided sequence is arithmetic.
- Because of this, the sequence is not arithmetic.

### Writing Terms of Arithmetic Sequences

After recognizing an arithmetic sequence, we can determine the terms if we are provided the first term as well as the common difference between the two terms. The terms may be discovered by starting with the first term and repeatedly adding the common difference to the end of the list. In addition, any term may be obtained by inserting the values of and into the formula below, which is shown below. Find the first many terms of an arithmetic series based on the first term and the common difference of the sequence.

- To determine the second term, add the common difference to the first term
- And so on. To determine the third term, add the common difference to the second term
- This will give you the third term. Make sure to keep going until you’ve found all of the needed keywords
- Create a list of words separated by commas and enclosed inside brackets

Creating Arithmetic Sequences in the Form of Terms Fill in the blanks with the first five terms of the arithmetic sequencewithand. Adding three equals the same as deleting three. Starting with the first phrase, remove 3 from each word to arrive at the next term in the sequence. The first five terms are as follows: Analysis As predicted, the graph of the series is made up of points on a line, as seen in the figure (Figure). List the first five terms of the arithmetic sequence beginning with and ending with and.

There are two ways to write the sequence: in terms of the first word, 8, and the common difference.

We can identify the commonalities between the two situations.

Analysis Observe how each term’s common difference is multiplied by one in order to identify the following terms: once to find the second term, twice to get the third term, and so on.

### Using Recursive Formulas for Arithmetic Sequences

Arithmetic Sequences in the Form of Terms Make a list of the first five terms of the arithmetic sequence with and without Subtracting 3 from any number is equivalent to adding 3. Starting with the first phrase, deduct 3 from each word to arrive at the next term in the progression. First and foremost, there are five terms: Analysis To be predicted, the sequence’s diagram is made up of points on a line, as seen in Figure 1. (Figure). Write down the first five terms of the arithmetic series that begins with and ends withand Arithmetic Sequences in the Form of Terms Find out what you’ve got.

In this case, we know that the fourth term equals 14 and that it has the form.

By adding the common difference to the fourth term, we may find the fifth term.

The tenth term might be obtained by multiplying the common difference by the first term nine times, or by using the following equation:

- To discover the common difference between two terms, subtract any phrase from the succeeding term. In the recursive formula for arithmetic sequences, start with the initial term and substitute the common difference

Making a Recursive Formula for an Arithmetic Sequence is a difficult task. Write a recursive formula for the arithmetic series in the following format: The first term is defined as follows. It is possible to calculate the common difference by subtracting the first term from the second term. In the recursive formula for arithmetic sequences, substitute the initial term and the common difference in place of the first term. Analysis We can observe that the common difference is the slope of the line generated when the terms in the sequence are graphed, as illustrated in the figure below (Figure).

Is it necessary to deduct the first term from the second term in order to obtain the common difference between the two?

We can take any phrase in the sequence and remove it from the term after it.

Create a recursive formula for the arithmetic sequence using the information provided.

### Using Explicit Formulas for Arithmetic Sequences

It is possible to think of anarithmetic sequence as a function on the domain of natural numbers; it is a linear function since the rate of change remains constant throughout the series. The constant rate of change, often known as the slope of the function, is the most frequently seen difference. If we know the slope and the vertical intercept of a linear function, we can create the function. For the -intercept of the function, we may take the common difference from the first term in the sequence and remove it from the result.

The common difference is that the series indicates a linear function with a slope of, whereas the difference is that We subtract from in order to determine the-intercept.

The graph is displayed in the following: (Figure).

The following equation is obtained by substituting for the slope as well as for the vertical intercept: To create an explicit formula for an arithmetic series, we do not need to identify the vertical intercept of the sequence.

- Identify the commonalities and differences
- Replace the common difference and the first word with the following:

After that, I’ll write the term paper. An Arithmetic Sequence with a Clearly Defined Formula Create an explicit formula for the arithmetic series using the following syntax: It is possible to calculate the common difference by subtracting the first term from the second term. The most often encountered difference is ten.

To simplify the formula, substitute the common difference and the first term in the series into it. Analysis It can be shown in (Figure) that the slope of this sequence is 10 and that the vertical intercept is 10. For the arithmetic series that follows, provide an explicit formula for it.

#### Finding the Number of Terms in a Finite Arithmetic Sequence

When determining the number of terms in a finite arithmetic sequence, explicit formulas can be employed to make the determination. Finding the common difference and determining the number of times the common difference must be added to the first term in order to produce the last term of the sequence are both necessary steps. The Number of Terms in a Finite Arithmetic Sequence can be determined by the following method: The number of terms in the infinite arithmetic sequence is to be determined.

The most noticeable change is.

substitute for and find a solution for There are a total of eight terms in the series.

There are a total of 11 terms included in the sequence.

#### Solving Application Problems with Arithmetic Sequences

When determining the number of terms in a finite arithmetic sequence, explicit formulae can be employed to aid in the calculation of the result. Finding the common difference and determining the number of times the common difference must be added to the first term in order to produce the last term of the sequence are both important tasks. In a Finite Arithmetic Sequence, finding the number of terms is a simple task. The number of terms in the infinite arithmetic sequence may be found by counting them.

The most noticeable distinction is.

Replace and solve for the following terms: Throughout the sequence, there are eight different words.

- Create a method for calculating the child’s weekly stipend over the course of a year
- What will be the child’s allowance when he reaches the age of sixteen

- Calculate the child’s weekly allowance using a formula for a certain year. Which amount will be the child’s allowance when he reaches the age of sixteen?

A lady chooses to go for a 10-minute run every day this week, with the goal of increasing the length of her daily exercise by 4 minutes each week after that. Create a formula to predict the timing of her run after n weeks has passed. In eight weeks, how long will her daily run last on average? The formula is, and it will take her 42 minutes to complete the task.

### Section Exercises

What is an arithmetic sequence, and how does it work? A series in which each subsequent term grows (or lowers) by a constant value is known as a constant value sequence. What is the procedure for determining the common difference of an arithmetic sequence? What is the best way to tell if a sequence is arithmetic or not? If the difference between all successive words is the same, we may say that they are sequential. This is the same as saying that the sequence has a common difference between it and the rest of the series.

What are the primary distinctions between the two methods?

What is the difference between them?

These two types of functions are distinct because their domains are not the same; linear functions are defined for all real numbers, whereas arithmetic sequences are specified for natural numbers or a subset of the natural numbers, respectively.

#### Algebraic

Work on finding the common difference for the arithmetic sequence that is supplied in the following problems. The most noticeable distinction is Determine if the sequence in the following exercises is mathematical or not in the following exercises. If this is the case, identify the common difference. Because of this, the sequence is not arithmetic. Write the first five terms of the arithmetic sequence given the first term and the common difference in the following tasks. Fill in the blanks with the first five terms of the arithmetic series given two terms in the following problems.

- The first term is 3, the most common difference is 4, and the fifth term is 3.
- The first term is 5, the most common difference is 6, and the eighth term is The first term is 6, the most common difference is 7, and the sixth term is 6.
- To complete the following problems, you must discover the first term from an arithmetic series given two terms.
- Make use of the recursive formula in order to write the first five terms of the arithmetic sequence in each of the following tasks.
- The following activities require you to build a recursive formula for the provided arithmetic sequence, followed by a search for the required term in the formula.
- Find the fourteenth term.
- For the following problems, write the first five terms of the arithmetic sequence using the explicit formula that was provided.
- The number of terms in the finite arithmetic sequence presented in the following task is to be determined for the following exercises: The series has a total of ten terms.

#### Graphical

Determine whether or not the graph given reflects an arithmetic sequence in the following tasks. The graph does not reflect an arithmetic sequence in the traditional sense. To complete the following activities, use the information supplied to graph the first 5 terms of the arithmetic sequence using the information provided.

#### Technology

In order to complete the following activities with an arithmetic sequence and a graphing calculator, follow the methods listed below:

- Select SEQ in the fourth line
- Select DOT in the fifth line
- Then press Enter or Return.

In the fourth line, choose SEQ; in the fifth line, choose DOT. Then press; In the column with the header “First Seven Terms,” what are the first seven terms listed? Use the scroll-down arrow to move to the next page. What is the monetary value assigned to Press. Set the parameters and then press. The sequence should be graphed exactly as it appears on the graphing calculator. Follow the techniques outlined above to work with the arithmetic sequence using a graphing calculator for the following problems.

Make careful to modify the WINDOW settings as necessary.

#### Extensions

Examples of arithmetic sequences with four terms that are the same are shown. Give two instances of arithmetic sequences whose tenth terms are as follows: There will be a range of responses. Examples: and The fifth term of the arithmetic series must be discovered Determine the eleventh term in the arithmetic series. At what point does the sequence reach the number 151? When does the series begin to contain negative values and at what point does it stop? For which terms does the finite arithmetic sequence have integer values?

Create an arithmetic series using a recursive formula to demonstrate your understanding.

There will be a range of responses.

Example: Formula for recursion: The first four terms are as follows: Create an arithmetic sequence using an explicit formula to demonstrate your understanding.

### Glossary

Arithmetic sequencea sequence in which the difference between any two consecutive terms is a constantcommon difference is a series in which the difference between any two consecutive terms is a constant an arithmetic series is the difference between any two consecutive words in the sequence