Solution: To find a specific term of an arithmetic sequence, we use the formula for finding the nth term. Step 1: The nth term of an arithmetic sequence is given by **an = a + (n – 1)d**. So, to find the nth term, substitute the given values a = 2 and d = 3 into the formula.

Contents

- 1 How do you calculate arithmetic sequences?
- 2 What is the formula of the sum of arithmetic sequence?
- 3 How do you find the 25th term of an arithmetic sequence?
- 4 What is example of arithmetic sequence?
- 5 What is the arithmetic mean between 16 and 30?
- 6 Formulas for Arithmetic Sequences
- 7 Using Explicit Formulas for Arithmetic Sequences
- 7.1 A General Note: Explicit Formula for an Arithmetic Sequence
- 7.2 How To: Given the first several terms for an arithmetic sequence, write an explicit formula.
- 7.3 Example: Writing then th Term Explicit Formula for an Arithmetic Sequence
- 7.4 Try It
- 7.5 A General Note: Recursive Formula for an Arithmetic Sequence
- 7.6 How To: Given an arithmetic sequence, write its recursive formula.
- 7.7 Example: Writing a Recursive Formula for an Arithmetic Sequence
- 7.8 How To: Do we have to subtract the first term from the second term to find the common difference?
- 7.9 Try It

- 8 Find the Number of Terms in an Arithmetic Sequence
- 9 Solving Application Problems with Arithmetic Sequences
- 10 Contribute!
- 11 Arithmetic Sequences and Sums
- 12 Arithmetic Sequence
- 13 Advanced Topic: Summing an Arithmetic Series
- 14 Footnote: Why Does the Formula Work?
- 15 Arithmetic Sequence Formula – What is Arithmetic Sequence Formula? Examples
- 16 What Is the Arithmetic Sequence Formula?
- 17 Applications of Arithmetic Sequence Formula
- 18 Examples Using Arithmetic Sequence Formula
- 19 FAQs on Arithmetic Sequence Formula
- 20 Arithmetic Sequences and Series
- 21 Arithmetic Series
- 22 Arithmetic Sequences
- 23 Arithmetic Sequences – Formula for n-th Term
- 24 FORMULA (for the n-th term)
- 25 EXAMPLE
- 26 Exercise 1
- 27 EXAM STYLE Questions
- 28 Arithmetic sequences calculator that shows work
- 29 About this calculator
- 30 Arithmetic & Geometric Sequences
- 30.0.1 Find the common difference and the next term of the following sequence:
- 30.0.2 Find the common ratio and the seventh term of the following sequence:
- 30.0.3 Find the tenth term and then-th term of the following sequence:
- 30.0.4 Find then-th term and the first three terms of the arithmetic sequence havinga6= 5andd=
- 30.0.5 Find then-th term and the first three terms of the arithmetic sequence havinga4= 93anda8= 65.
- 30.0.6 Find then-th and the26 th terms of the geometric sequence withanda12= 160.

- 31 Arithmetic Sequence Calculator
- 32 What is an arithmetic sequence?
- 33 Arithmetic sequence definition and naming
- 34 Arithmetic sequence examples
- 35 Arithmetic sequence formula
- 36 Difference between sequence and series
- 37 Arithmetic series to infinity
- 38 Arithmetic and geometric sequences
- 39 Arithmetico–geometric sequence
- 40 Arithmetic sequence calculator: an example of use

## How do you calculate arithmetic sequences?

An arithmetic sequence is a sequence in which the difference between each consecutive term is constant. An arithmetic sequence can be defined by an explicit formula in which a_{n} = d (n – 1) + c, where d is the common difference between consecutive terms, and c = a_{1}.

## What is the formula of the sum of arithmetic sequence?

The sum of the arithmetic sequence can be derived using the general arithmetic sequence, an n = a1 1 + (n – 1)d.

## How do you find the 25th term of an arithmetic sequence?

Solution: A sequence in which the difference between all pairs of consecutive numbers is equal is called an arithmetic progression. The sequence given is 3, 9, 15, 21, 27, … Therefore, the 25th term is 147.

## What is example of 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. If we add or subtract by the same number each time to make the sequence, it is an arithmetic sequence.

## What is the arithmetic mean between 16 and 30?

Find the arithmetic mean of 16 and 30. Solution: Here a=16 and b=30. Arithmetic mean = (a+b)/2 = (16+30)/2 = 46/2 = 23.

## Formulas for Arithmetic Sequences

- Create a formal formula for an arithmetic series using explicit notation
- Create a recursive formula for the arithmetic series using the following steps:

## 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. = +dleft = +dright 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.

Considering that the average difference is 50, the series represents a linear function with an associated slope of 50.

You may also get the they-intercept by graphing the function and calculating the point at which a line connecting the points would intersect the vertical axis, as shown in the example.

When working with sequences, we substitute _instead of y and ninstead of n.

Using 50 as the slope and 250 as the vertical intercept, we arrive at this equation: = -50n plus 250 To create an explicit formula for an arithmetic series, we do not need to identify the vertical intercept of the sequence.

### A General Note: Explicit Formula for an Arithmetic Sequence

For the textterm of an arithmetic sequence, the formula = +dleft can be used to express it explicitly.

### How To: Given the first several terms for an arithmetic sequence, write an explicit formula.

- Find the common difference between the two sentences, – To solve for = +dleft(n – 1right), substitute the common difference and the first term into the equation

### Example: Writing then th Term Explicit Formula for an Arithmetic Sequence

Create an explicit formula for the arithmetic sequence.left 12text 22text 32text 42text ldots right 12text 22text 32text 42text ldots

### Try It

For the arithmetic series that follows, provide an explicit formula for it. left With the use of a recursive formula, several arithmetic sequences may be defined in terms of the preceding term. The formula contains an algebraic procedure that may be used to determine the terms of the series. We can discover the next term in an arithmetic sequence by utilizing a function of the term that came before it using a recursive formula. In each term, the previous term is multiplied by the common difference, and so on.

The initial term in every recursive formula must be specified, just as it is with any other formula.

### A General Note: Recursive Formula for an Arithmetic Sequence

In the case of an arithmetic sequence with common differenced, the recursive formula is as follows: the beginning of the sentence = +dnge 2 the finish of the sentence

### How To: Given an arithmetic sequence, write its recursive formula.

- 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

### Example: Writing a Recursive Formula for an Arithmetic Sequence

Write a recursive formula for the arithmetic series in the following format: left

### How To: Do we have to subtract the first term from the second term to find the common difference?

No.

We can take any phrase in the sequence and remove it from the term after it. Generally speaking, though, it is more customary to subtract the first from the second term since it is frequently the quickest and most straightforward technique of determining the common difference.

### Try It

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

## Find the Number of Terms in an 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.

### How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

- Find the common differences between the two
- To solve for = +dleft(n – 1right), substitute the common difference and the first term into the equation Fill in the blanks with the final word from and solve forn

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

The number of terms in the infinite arithmetic sequence is to be determined. left

### Try It

The number of terms in the infinite arithmetic sequence may be found by counting them. left

## Solving Application Problems with Arithmetic Sequences

In many application difficulties, it is frequently preferable to begin with the term instead of_ as an introductory phrase. When solving these problems, we make a little modification to the explicit formula to account for the change in beginning terms. The following is the formula that we use: = +dn = = +dn

### Example: Solving Application Problems with Arithmetic Sequences

Every week, a kid under the age of five receives a $1 stipend from his or her parents. His parents had promised him a $2 per week rise on a yearly basis.

- 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

### Try It

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?

## Contribute!

Do you have any suggestions about how to make this article better? We would much appreciate your feedback. Make this page more user-friendly. Read on to find out more

## 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:

## Arithmetic Sequence Formula – What is Arithmetic Sequence Formula? Examples

Calculating the nth term of an arithmetic progression is accomplished through the use of the arithmetic sequence formula. The arithmetic sequence is a series in which the common difference between any two succeeding terms remains constant throughout the sequence. In order to discover any term in the arithmetic sequence, we may use the arithmetic sequence formula, which is defined as follows: Let’s look at several solved cases to better grasp the arithmetic sequence formula.

## What Is the Arithmetic Sequence Formula?

An Arithmetic sequence has the following structure: a, a+d, a+2d, a+3d, and so on up to n terms. In this equation, the first term is called a, the common difference is called d, and n = the number of terms is written as n. Recognize the arithmetic sequence formulae and determine the AP, first term, number of terms, and common difference before proceeding with the computation. Various formulae linked with an arithmetic series are used to compute the n thterm, total, or common difference of a given arithmetic sequence, depending on the series in question.

### Arithmetic Sequence Formula

The arithmetic sequence formula is denoted by the notation Formula 1 is a racing series that takes place on the track. The arithmetic sequence formula is written as (a_ =a_ +(n-1) d), where an is the number of elements in the series.

- A_ is the n th term
- A_ is the initial term
- And d is the common difference.

The n thterm formula of anarithmetic sequence is sometimes known as the n thterm formula of anarithmetic sequence.

For the sum of the first n terms in an arithmetic series, the formula is (S_ = frac), where S is the number of terms.

- (S_ ) is the sum of n terms
- (S_ ) is the sum of n terms
- A is the initial term, and d is the difference between the following words that is common to all of them.

Formula 3: The formula for determining the common difference of an AP is given as (d=a_ -a_ )where, a_ is the AP’s initial value and a_ is the common difference of the AP.

- There are three terms in this equation: nth term, second last term, and common difference between the consecutive terms, denoted by the letter d.

Formula 4: When the first and last terms of an arithmetic progression are known, the sum of the first n terms of the progression is given as, (s_ = fracleft )where, and

- (S_ ) is the sum of the first n terms
- (a_ ) is the last term
- And (a_ ) is the first term.

## Applications of Arithmetic Sequence Formula

Each and every day, and sometimes even every minute, we employ the arithmetic sequence formula without even recognizing it. The following are some examples of real-world uses of the arithmetic sequence formula.

- Arranging the cups, seats, bowls, or a house of cards in a towering fashion
- There are seats in a stadium or a theatre that are set up in Arithmetic order
- The seconds hand on the clock moves in Arithmetic Sequence, as do the minutes hand and the hour hand
- The minutes hand and the hour hand also move in Arithmetic Sequence. The weeks in a month follow the AP, and the years follow the AP as well. It is possible to calculate the number of leap years simply adding 4 to the preceding leap year. Every year, the number of candles blown on a birthday grows in accordance with the mathematical sequence

Consider the following instances that have been solved to have a better understanding of the arithmetic sequence formula. Do you want to obtain complicated math solutions in a matter of seconds? To get answers to difficult queries, you may use our free online calculator. Find solutions in a few quick and straightforward steps using Cuemath. Schedule a No-Obligation Trial Class.

## Examples Using Arithmetic Sequence Formula

In the first example, using the arithmetic sequence formula, identify the thirteenth term in the series 1, 5, 9, and 13. Solution: To locate the thirteenth phrase in the provided sequence. Due to the fact that the difference between consecutive terms is the same, the above sequence is an arithmetic series. a = 1, d = 4, etc. Making use of the arithmetic sequence formula (a_ =a_ +(n-1) d) = (a_ =a_ +(n-1) d) = (a_ =a_ +(n-1) d) For the thirteenth term, n = 13(a_ ) = 1 + (13 – 1) 4(a_ ) = 1 + 4(a_ ) (12) The sum of 4(a_ ) and 48(a_ ) equals 49.

- Example 2: Determine the first term in the arithmetic sequence in which the 35th term is 687 and the common difference between the two terms.
- Solution: In order to locate: The first term in the arithmetic sequence is called the initial term.
- Example 3: Calculate the total of the first 25 terms in the following sequence: 3, 7, 11, and so on.
- In this case, (a_ ) = 3, d = 4, n = 25.
- With the help of the Sum of Arithmetic Sequence Formula (S_ =frac), we can calculate the sum of the first 25 terms (S_ =frac) as follows: (25/2) = 25/2 102= 1275.

## FAQs on Arithmetic Sequence Formula

It is referred to as arithmetic sequence formula when it is used to compute the general term of an arithmetic sequence as well as the sum of all n terms inside an arithmetic sequence.

### What Is n in Arithmetic Sequence Formula?

It is important to note that in the arithmetic sequence formula used to obtain the generalterm (a_ =a_ +(n-1) d), n refers to how many terms are in the provided arithmetic sequence.

### What Is the Arithmetic Sequence Formula for the Sum of n Terms?

The sum of the first n terms in an arithmetic series is denoted by the expression (S_ =frac), where (S_ ) =Sum of n terms, (a_ ) = first term, and (d) = difference between the first and second terms.

### How To Use the Arithmetic Sequence Formula?

The sum of the first n terms in an arithmetic series is denoted by the expression (S_ =frac), where (S_ ) =Sum of n terms, (a_ ) = first term, and (d) = difference between the first and last terms.

- This is the formula for thearithmetic sequence: (a_ =a_ +(n-1) d), where a_ is a general term, a_ is a first term, and d is the common difference between the two terms. This is done in order to locate the general word inside the sequence. The sum of the first n terms in an arithmetic series is denoted by the symbol (S_ =frac), where (S_ ) =Sum of n terms, (a_ )=first term, and (d) represents the common difference between the terms. When computing the common difference of an arithmetic series, the formula is stated as, (d=a_ -a_ ), where a_ is the nth term, a_ is the second last term, and d is the common difference. Arithmetic progression is defined as follows: (s_ =fracleft) = Sum of first n terms, nth term, and nth term
- (s_ =fracright) = First term
- (s_ =fracleft)= Sum of first two terms
- And (s_ =fracright) = Sum of first n terms.

## 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.

The terms of an arithmetic sequence that occur between two supplied terms.

### Example 3

Find all of the words that fall between a1=8 and a7=10. in the context of an arithmetic series Or, to put it another way, locate all of the arithmetic means between the 1st and 7th terms. Solution: Begin by identifying the points of commonality. In this situation, we are provided with the first and seventh terms, respectively: an=a1+(n−1) d Make use of n=7.a7=a1+(71)da7=a1+6da7=a1+6d Substitutea1=−8anda7=10 into the preceding equation, and then solve for the common differenced result. 10=−8+6d18=6d3=d Following that, utilize the first terma1=8.

a1=3(1)−11=3−11=−8a2=3 (2)−11=6−11=−5a3=3 (3)−11=9−11=−2a4=3 (4)−11=12−11=1a5=3 (5)−11=15−11=4a6=3 (6)−11=18−11=7} In arithmetic, a7=3(7)11=21=10 means a7=3(7)11=10 Answer: 5, 2, 1, 4, 7, and 8.

### 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!

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. There are two variables, a1 and a100. 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.
- Calculate the sum of the first 60 terms of the following sequence of numbers: 5, 0, 5, 10, 15,.
- 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

- Make a calculation for the provided total based on the formula for the general term an=3n+5
- S100
- An=5n11
- An=12n
- S70
- An=132n
- S120
- An=12n34
- S20
- An=n35
- S150
- An=455n
- S65
- An=2n48
- S95
- An=4.41.6n
- S75
- An=6.5n3.3
- S67
- An=3n+5

- 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

- It may be expressed asan=2n1 where an=2n1 is the generic term for the sequence of positive odd numbers. in this case, byan=2n is used to denote a generic phrase for a succession of positive even numbers. The following are to be found
- There are four different types of sums: 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
- And the sum of the first 200 positive even integers. The sum of the firstk positive odd integers
- The sum of the firstk positive odd integers (a) The sum of the firstkpositive even integers
- (b) The sum of the firstknegative even integers
- In a tiny theater, the first row of seating has a total of 8 places. Following that, each row has three additional seats than the one before it. How many total seats are there in the theater if there are 12 rows? If you’re looking for seats in an outdoor amphitheater, the first row has 42 seats, the second row has 44 seats, the third row has 46 seats, and so on. What is the overall seating capacity of the theater if there are 22 rows of seats
- And 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 pile? Each succeeding row of a triangle stack of bricks contains one fewer brick, until there is just one brick remaining on the summit of the pile. If there are 210 bricks in all, how many rows does the stack have? a wage contract with a 10-year term that pays $65,000 in the first year and increases by $3,200 every year afterwards Calculate the entire salary obligation over the course of ten years. It is customary for a clock tower to chime its bell a certain number of times every hour. The clock strikes once at one o’clock, twice at two o’clock, and so on throughout the day. I’m curious how many times the clock tower’s bell rings in a day.

### 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

- 2,450, 90, 7,800, 4,230, 38,640, 124,750, 18,550, 765, 10,000, 20,100, 2,500, 2,550, K2, 294 seats, 247 bricks, $794,000, and

## 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 Sequences – Formula for n-th Term

Whenever we move from one term to the next in an anarithmetic sequence, also known as linear sequence, we add the same amount to each term. When we sum up all of the amounts, we call it the “common difference,” and we refer to it by the letter “d.” For example, consider the following instances of arithmetic sequences:

- An arithmetic sequence with common difference d = 5
- An arithmetic sequence with common difference d = -3
- An arithmetic sequence with common difference d = -1.5

### Introduction to Arithmetic SequencesFormula for n-th term

In this lesson, we will learn about arithmetic sequences as well as the formula for the n-th term in a sequence.

## FORMULA (for the n-th term)

The following formula may be used to compute any of the terms in an arithmetic sequence given the following: where:

- The first phrase in the series is denoted by the letter u 1. The common difference is represented by the letter (d).

## EXAMPLE

Given the sequence of integers in the following example:

- Identify its n-th term’s formula (which is the formula for calculating any term)
- Figure out what the tenth term is.

### Solution

Identify its n-th term’s formula (which may be used to calculate any term). Figure out what the tenth term is worth;

- The first term, which is (u 1 = 3)
- The common difference, which is (d = 4)
- And the second term, which is (u 2 = 3).

Substituting both u 1 and d in the formula (u n = u 1 + begin n – 1end.d) results in the following result: In order to compute any term of the sequence, we must first substitute every n that appears in the formula with the number of the term that we desire to discover. For example, if we wish to compute the tenth term, we would substitute every n that we see with the number ten, as seen in this equation: Beginu = 3 + Begin10- 1 end. [Beginu = 3 + Begin10- 1 end. 4 times 3 times 9 times 3 times 9 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 times 9 times 3 times 3 [4 = 3 + 36 u = 39] [4= 3 + 36 u = 39] [4= 3 + 36 u = 39] [4= 3 + 36 u = 39] [4= 3 + 36 u = 39] [4= 3 + 36 u = 39] [4= 3 + 36 u = 39] [4= 3 + 36 u = 39] [4= 3 + 36 u = 39] [4= 3 +

### Arithmetic or Linear Sequences

The following is another way to write the formula for the n-th term of an arithmetic sequence: where (c) is a constant. (In reality, (c = u 1 – d) is true). When we start with (u n = u 1 + n-1end.d) and then extend the parenthesis, we can readily see that (c = u 1-d). Indeed: [beginu n= u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + dn – d u n=dn + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u_

### Example

The following is another way of writing the formula for the n-th term of an arithmetic sequence: where (c) is a constant. As a result, (c = u 1 d) is true. From the expression (u n = u 1 + n-1 + end.d) and widening the parenthesis, we can readily determine that c = (u 1-d). [[beginu n= u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + u 1 + dn + u n=dn+ underbrace _end ][[beginu n= u 1 + u 1 + u 1 + u 1 – u 1

### Tutorial

Any arithmetic sequence may be expressed as alinear sequence in this tutorial, which will be covered in detail in the next session.

### Common Difference Formula

We can always find the common difference given an arithmetic series by using the following formula: which may also be written:

### Explanation

By subtracting each word from the next, we may calculate the common difference d given an arithmetic sequence, as shown in the following formula: Using the following examples, given the arithmetic sequence: we may compute (d) using any of the following methods:

- By subtracting each word from the next, we may calculate the common difference d given an arithmetic sequence, as shown in the following formula. Using the following examples, given the arithmetic sequence: we may compute (d) using one of the following methods:

## Exercise 1

Perform each of the following actions for each of the arithmetic sequences, starting with a and ending with e:

- Specify the value of (u 1) in words. Indicator (d) is the value of the common difference. Find the formula for the n-th phrase, written in both English and latin characters:

- (9, 16, 23, 30, 37, dots beginu end )
- (32, 27, 22, 17, 12, dots beginu end)
- (41, 39, 37, 35, 33, dots beginu end )

## EXAM STYLE Questions

Following that, we will learn how to solve some typicalexam-style problems that will use arithmetic sequences as well as the formula for the n-thirteenth term. Begin by viewing the instructions that are provided here.

### Exam Question 1

There are five terms in an arithmetic series, with the first term equal to (5), and the last term equal to (5). (54). Find the common difference between these two sequences (d). The complete solution to this issue may be found in the tutorial that is linked to here on this page.

### Exam Question 1

The third term in an arithmetic series is equal to (4), and the seventh term is equivalent to (7). (22). Find the initial term (u 1) in this series, as well as the common difference (u 2) (d). The complete solution to this issue may be found in the tutorial that is linked to here on this page.

### Exam Question 2

- The first term of an arithmetic series is equal to (-5), and the fifth term is equal to (-5). (7). Find the common difference between these two sequences. The fifth term in an arithmetic series is equal to (1), and the ninth term is equivalent to (9). (-7). Find the initial term in this series, as well as the common difference between it. One example of an arithmetic progression is one in which the tenth term is equal to (8), and the second term is equal to (2). (4). Find out what its initial phrase is and what its common difference is
- The first term in an arithmetic series is (u 1 = 11) and the seventh term is (u 7 = -13) in the sequence. Determine the common difference (d) in this series. When given an arithmetic sequence in which the 5th term equals (10) and the 20th term equals (13), identify the sequence’s initial term and the common difference between the two terms. If you have an arithmetic progression whose first term is (u 1 = -1), and the fifth term is (-29), identify the common difference between these two terms. An arithmetic series has a 6th term equal to (19), and a 3rd term equal to (21), respectively (1). Find the first term in this sequence, as well as the difference between it and the preceding term. An arithmetic sequence has a 10th term equal to (53), and a 4th term equal to (53). An arithmetic sequence has a 10th term equal to (53). (17). Find out what its initial phrase is and what its common difference is

Please keep in mind that you may download this activity as a worksheet to use for practice: Worksheet number one

## Arithmetic sequences calculator that shows work

This online tool can assist you in determining the first $n$ term of an arithmetic progression as well as the total of the first $n$ terms of the progression. This calculator may also be used to answer even more complex issues than the ones listed above. For example, if $a 5 = 19 $ and $S 7 = 105$, the calculator may calculate the common difference ($d$) between the two numbers. Probably the most significant advantage of this calculator is that it will create all of the work with a thorough explanation.

- + 98 + 99 + 100 =?
- In an arithmetic series, the first term is equal to $frac$, and the common difference is equal to 2.
- An arithmetic series has a common difference of $7$ and its eighth term is equal to $43$, with the common difference being $7$.
- Suppose $a 3 = 12$ and the sum of the first six terms is equal to 42.
- When the initial term of an arithmetic progression is $-12$, and the common difference is $3$, then the progression is complete.

## About this calculator

This online tool can assist you in determining the first $n$ term of an arithmetic progression as well as the sum of the first $n$ terms. As an added bonus, you may use this calculator for even more difficult tasks. As an example, if $a 5 = 19 $ and $S 7 = 105$, the calculator may calculate the common difference ($d$). That it will create all of the work with thorough explanations is the calculator’s most significant advantage. examples A sum of one dollar, two dollars, three dollars, four dollars,.

Two terms in an arithmetic series are $frac$, and the common difference between them is equal to two.

There is a common difference of $7$ in an arithmetic series, and the 8th term is equivalent to $43$ in that sequence.

For example, if $a 3 = 12$ and $the sum of the first 6 terms is 42$, find the first term and difference of an arithmetic progression.

It is $-12$ for the first term of an arithmetic progression, and $3$ for the common difference of two terms of the same progression. To arrive at a total of $1104$, you must first decide how many phrases must be combined together.

## Arithmetic & Geometric Sequences

The arithmetic and geometric sequences are the two most straightforward types of sequences to work with. An arithmetic sequence progresses from one term to the next by adding (or removing) the same value on each successive term. For example, the numbers 2, 5, 8, 11, 14,.are arithmetic because each step adds three; while the numbers 7, 3, –1, –5,.are arithmetic because each step subtracts four. The number that is added (or subtracted) at each stage of an arithmetic sequence is referred to as the “common difference”d because if you subtract (that is, if you determine the difference of) subsequent terms, you will always receive this common value as a result of the process.

Below In a geometric sequence, the terms are connected to one another by always multiplying (or dividing) by the same value.

Each step of a geometric sequence is represented by a number that has been multiplied (or divided), which is referred to as the “common ratio.” If you divide (that is, if you determine the ratio of) subsequent terms, you’ll always receive this common value.

#### Find the common difference and the next term of the following sequence:

The arithmetic and geometric sequences are the two basic types of sequences to work with. It is possible to go from one term to another by adding (or deleting) the same value on each successive term. 2. 5. 8. 11. 14.is arithmetic because each step adds three; and 7. 3. –1, 5.is arithmetic because each step subtracts four; for example, 2, 5, 8, 11, 14. When a number is added (or subtracted) at each stage of an arithmetic sequence, it is referred to as the “common difference”d. This is because if you subtract (that is, if you discover the difference of) subsequent terms, you will always receive the same common value.

The numbers 1, 2, 4, 8, 16, and 81, 27, 9, 3, 1,.

are geometric because each step divides by three.

#### Find the common ratio and the seventh term of the following sequence:

To get the common ratio, I must divide each succeeding pair of terms by the number of terms in the series. There’s no point in choosing which couple I want to sit with as long as they’re right next to one other. I’ll go over all of the divisions to be thorough: The ratio is always three, hence sor= three. As a result, I have five terms remaining; the sixth term will be the next term, and the seventh will be the term after that. The value of the seventh term will be determined by multiplying the fifth term by the common ratio two times.

When it comes to arithmetic sequences, the common difference isd, and the first terma1is commonly referred to as “a “.

As a result of this pattern, the then-th terma n will take the form: n=a+ (n– 1)d When it comes to geometric sequences, the typical ratio isr, and the first terma1 is commonly referred to as “a “.

This pattern will be followed by a phrase with the following form: a n=ar(n– 1) is equal to a n. Before the next test, make a note of the formulae for the tenth term.

#### Find the tenth term and then-th term of the following sequence:

, 1, 2, 4, 8, and so forth. Identifying whether sort of sequence this is (arithmetic or geometric) is the first step in solving the problem. As soon as I look at the differences, I see that they are not equal; for example, the difference between the second and first terms is 2 – 1 = 1, while the difference between the third and second terms is 4 – 2 = 2. As a result, this isn’t a logical sequence. As an alternative, the ratios of succeeding terms remain constant. For example, Two plus one equals twenty-four plus two equals twenty-eight plus four equals two.

The division, on the other hand, would have produced the exact same result.) The series has a common ratio of 2 and the first term is a.

I can simply insert the following into the formulaa n=ar(n– 1) to obtain the then-th term: So, for example, I may plugn= 10 into the then-th term formula and simplify it as follows_n= 10 Then here’s what I’d say: n-th term: tenth term: 256 n-th term

#### Find then-th term and the first three terms of the arithmetic sequence havinga6= 5andd=

The n-th term in an arithmetic series has the form n=a+ (n– 1) d, which stands for n=a+ (n– 1) d. In this particular instance, that formula results in me. When I solve this formula for the value of the first term in the sequence, I obtain the resulta= Then:I have the first three terms in the series as a result of this. Because I know the value of the first term and the common difference, I can also develop the expression for the then-th term, which will be easier to remember: In such case, my response is as follows:n-th word, first three terms:

#### Find then-th term and the first three terms of the arithmetic sequence havinga4= 93anda8= 65.

Due to the fact thata4 anda8 are four places apart, I can determine from the definition of an arithmetic sequence that I can go from the fourth term to the eighth term by multiplying the common difference by four times the fourth term; in other words, the definition informs me that a8=a4 + 4 d. I can then use this information to solve for the common differenced: 65 = 93 + 4 d –28 = 4 d –7 = 65 = 93 + 4 d Also, I know that the fourth term is related to the first term by the formulaa4=a+ (4 – 1) d, so I can get the value of the first terma by using the value I just obtained ford and the value I just discovered fora: 93 =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) 93 plus 21 equals 114.

As soon as I know what the first term’s value is and what the value of the common difference is, I can use the plug-and-chug method to figure out what the first three terms’ values are, as well as the general form of the fourth term: The numbers are as follows: a1= 114, a2= 114– 7, a3= 107– 7, and an= 114 + (n – 1)(–7)= 114 – 7, n+ 7, and an= 121–7, respectively.

#### Find then-th and the26 th terms of the geometric sequence withanda12= 160.

Given that the two words for which they’ve provided numerical values are separated by 12 – 5 = 7 places, I know that I can go from the fifth term to the twelfth term by multiplying the fifth term by the common ratio seven times; that is, a12= (a5) (r7). I can use this to figure out what the value of the common ratior should be: I also know that the fifth component is related to the first by the formulaa5=ar4, so I can use that knowledge to solve for the value of the first term, which is as follows: Now that I know the value of the first term as well as the value of the common ratio, I can put both into the formula for the then-th term to obtain the following result: I can assess the twenty-sixth term using this formula, and it is as follows, simplified: Then here’s my response:n-th term: 2,621,440 for the 26th term Once we have mastered the art of working with sequences of arithmetic and geometric expressions, we may move on to the concerns of combining these sequences together.

## Arithmetic Sequence Calculator

Using this arithmetic sequence calculator (sometimes referred to as the arithmetic series calculator) you can easily analyze any sequence of integers that is generated by adding a constant value to each number in the sequence each time. You may use it to determine any attribute of a series, such as the first term, the common difference, the nth term, or the sum of the first n terms, among other possibilities. You may either start using it right away or continue reading to learn more about how it works.

An introduction of the distinctions between arithmetic and geometric sequences, as well as an easily understandable example of how to use our tool, are also included.

## What is an arithmetic sequence?

To answer this question, you must first understand what the terms sequence and sequencemean. In mathematics, a sequence is defined as a collection of items, such as numbers or characters, that are presented in a specified order, as defined by the definition. The items in this sequence are referred to as elements or terms of the sequence. It is fairly typical for the same object to appear more than once in a single sequence of pictures. An arithmetic sequence is also a collection of items — in this case, a collection of numbers.

Such a series can be finite if it contains a certain number of terms (for example, 20 phrases), or it can be unlimited if we do not define the number of words to be contained.

If you know these two numbers, you’ll be able to write out the entire sequence in your head.

## Arithmetic sequence definition and naming

The concept of what is an arithmetic sequence may likely cause some confusion when you first start looking into it, so be prepared for that. It occurs as a result of the many name standards that are now in use. The words arithmetic sequence and series are two of the most often used terms in mathematics. The first of them is also referred to as anarithmetic progression, while the second is referred to as the partial sum. When comparing sequence and series, the most important distinction to note is that, by definition, an arithmetic sequence is just the set of integers generated by adding the common difference each time.

For example, S 12= a 1+ a 2+.

## Arithmetic sequence examples

The following are some instances of an arithmetic sequence:

- 3, 5, 7, 9, 11, 13, 15, 17, 19, 21,.
- 6, 3, 0, -3, -6, -9, -12, -15,.
- 50, 50.1, 50.2, 50.3, 50.4, 50.5,.

Is it possible to identify the common difference between each of these sequences? As a hint, try deleting a term from the phrase after this one. You can see from these examples of arithmetic sequences that the common difference does not necessarily have to be a natural number; it may be a fraction instead. In fact, it isn’t even necessary that it be favorable! In arithmetic sequences, if the common difference between them is positive, we refer to them as rising sequences. The series will naturally be descending if the difference between the two numbers is negative.

- As a result, you will have a amonotone sequence, in which each term is the same as the one before.
- are all possible combinations of numbers.
- You shouldn’t be allowed to do so in any case.
- Each phrase is discovered by adding the two terms that came before it.

A fantastic example of the Fibonacci sequence in action is the construction of a spiral. If you drew squares with sides that were the same length as the consecutive terms of this sequence, you’d have a perfect spiral as a result. This spiral is a beautiful example of perfection! (credit:Wikimedia)

## Arithmetic sequence formula

Consider the scenario in which you need to locate the 30th term in any of the sequences shown above (except for the Fibonacci sequence, of course). It would be hard and time-consuming to jot down the first 30 terms in this list. The good news is that you don’t have to write them all down, as you presumably already realized! If you add 29 common differences to the first term, that is plenty. Let’s generalize this assertion to produce the arithmetic sequence equation, which can be written as It is the formula for any nth term in a sequence that is not a prime.

- A1 is the first term of the series
- An is the nth phrase of the sequence
- D is the common difference
- And A is the nth term of the sequence

Whether the common differences are positive, negative, or equal to zero, this arithmetic sequence formula may be used to solve any problem involving arithmetic sequences. It goes without saying that in the event of a zero difference, all terms are equal to one another, making any computations redundant.

## Difference between sequence and series

For your convenience, our arithmetic sequence calculator can also calculate the sum of the sequence (also known as the arithmeticseries). Believe us when we say that you can do it yourself – it isn’t that difficult! Take a look at the first example of an arithmetic sequence: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 in the number 3. We could do a manual tally of all of the words, but this is not essential. Let’s try to structure the terms in a more logical way to summarize them. First, we’ll combine the first and last terms, followed by the second and second-to-last terms, third and third-to-last terms, and so on.

This implies that we don’t have to add up all of the numbers individually.

This is represented as S = n/2 * (a1 + a) in mathematical terms.

## Arithmetic series to infinity

When attempting to find the sum of an arithmetic series, you have surely observed that you must choose the value ofn in order to compute the partial sum of the sequence. What if you wanted to condense all of the terms in the sequence into one sentence? With the right intuition, the sum of an infinite number of terms will equal infinity, regardless of whether the common difference is positive, negative, or even equal to zero in magnitude. However, this is not always the true for all sorts of sequences.

## Arithmetic and geometric sequences

No other form of sequence can be analysed by our arithmetic sequence calculator, which should come as no surprise. For example, there is no common difference between the numbers 2, 4, 8, 16, 32,., and the number 2. This is due to the fact that it is a distinct type of sequence — ageometric progression. When it comes to sequences, what is the primary distinction between an algebraic and a geometric sequence? While an arithmetic sequence constructs each successive phrase using a common difference, a geometric sequence constructs each consecutive term using a common ratio.

The so-called digital universe is an interesting example of a geometric sequence that is worth exploring.

You’ve probably heard that the amount of digital information doubles in size every two years, and this is correct. Essentially, it implies that you may create a geometric series of integers expressing the quantity of data in which the common ratio is two in order to convey the amount of data.

## Arithmetico–geometric sequence

A unique sort of sequence, known as a thearithmetico-geometric sequence, may also be studied in detail. In order to produce it, you must multiply the terms of two progressions: an arithmetic progression and a geometric progression. Think about the following two progressions, as an illustration:

- The arithmetic series is as follows: 1, 2, 3, 4, 5,.
- The geometric sequence is as follows: 1, 2, 4, 8, 16,.

1, 2, 3, 4, 5,.; Arithmetic sequence: 1, 2, 3, 4, 5,.; Geometric sequence: 1, 2, 4, 8, 16,.; and more sequences.

- The first term is 1 * 1
- The second term is 2 * 2
- The third term is 3 * 4
- The fourth term is 4 * 8
- And the fifth term is 5 * 16 = 80.

Four parameters define such a sequence: the initial value of the arithmetic progressiona, the common differenced, the initial value of the geometric progressionb, and the common ratior. These parameters are described as follows:

## Arithmetic sequence calculator: an example of use

Let’s look at a small scenario that can be solved using the arithmetic sequence formula and see what we can learn. We’ll take a detailed look at the free fall scenario as an example. A stone is tumbling freely down a deep pit of darkness. Four meters are traveled in the first second of the video game’s playback. Every second that passes, the distance it travels increases by 9.8 meters. What is the distance that the stone has traveled between the fifth and ninth seconds of the clock? It is possible to plot the distance traveled as an arithmetic progression, with an initial value of 4 and a common difference of 9.8 meters.

However, we are only concerned with the distance traveled from the fifth to the ninth second of the second.

Simply remove the distance traveled in the first four seconds (S4,) from the partial total S9.

S4 = n/2 *= 4/2 *= 74.8 m = n/2 *= 4/2 *= S4 is the same as 74.8 meters.

It is possible to use the arithmetic sequence formula to compute the distance traveled in each of the five following seconds: the fifth, sixth, seventh, eighth, and ninth seconds.

Make an attempt to do it yourself; you will quickly learn that the outcome is precisely the same!