The arithmetic sequence formula is given as, **an=a1+(n−1)d a n = a 1 + ( n − 1 ) d** where, an a n = a general term, a1 a 1 = first term, and and d is the common difference. This is to find the general term in the sequence.

Contents

- 1 How do you find the arithmetic sequence?
- 2 What is the arithmetic mean between 10 and 24?
- 3 How do you find the sum of n terms in an arithmetic sequence?
- 4 What is the arithmetic mean between 19 and 7?
- 5 What is the arithmetic mean between 10 and 20?
- 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 Calculator
- 16 What is an arithmetic sequence?
- 17 Arithmetic sequence definition and naming
- 18 Arithmetic sequence examples
- 19 Arithmetic sequence formula
- 20 Difference between sequence and series
- 21 Arithmetic series to infinity
- 22 Arithmetic and geometric sequences
- 23 Arithmetico–geometric sequence
- 24 Arithmetic sequence calculator: an example of use
- 25 Arithmetic Sequences and Series
- 26 Arithmetic Series
- 27 Arithmetic sequences calculator that shows work
- 28 About this calculator
- 29 Number Sequence Calculator
- 30 Geometric Sequence Calculator
- 31 Fibonacci Sequence Calculator
- 32 How to Find Any Term of an Arithmetic Sequence
- 33 Video
- 34 About This Article
- 35 Did this article help you?

## 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.

## What is the arithmetic mean between 10 and 24?

Using the average formula, get the arithmetic mean of 10 and 24. Thus, 10+24/2 =17 is the arithmetic mean.

## How do you find the sum of n terms in an arithmetic sequence?

We use the first term (a), the common difference (d), and the total number of terms (n) in the AP to find its sum. The formula used to find the sum of n terms of an arithmetic sequence is n/2 (2a+(n−1)d).

## What is the arithmetic mean between 19 and 7?

Solution:Arithmetic mean between 7 and 19 is 13.

## What is the arithmetic mean between 10 and 20?

An arithmetic mean is a fancy term for what most people call an “average.” When someone says the average of 10 and 20 is 15, they are referring to the arithmetic mean. Then divide by 3 because we have three values, and we get an arithmetic mean (average) of 19.

## Formulas for Arithmetic Sequences

- Modular arithmetic is widely employed in pure mathematics, and it is considered to be a cornerstone of number theory. However, it also has a wide range of practical uses. It is used to create checksums for international standard book numbers (ISBNs) and bank IDs (Iban numbers), as well as to detect problems in these numbers. Modern business is dependent on public key cryptography technologies, which are based on modular arithmetic. It is also commonly used in the field of computer science. Finally, in music theory, modulo 12 arithmetic is used to analyze the 12-tone equal temperament system, in which sounds separated by an octave of 12 semitones are viewed as comparable.

## 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 finite arithmetic sequence has to be determined. 11 text 16 text. text 56 right 11 text 16 text 16 text 56 text 56 text 56 Following that, we’ll go over some of the concepts that have been introduced so far concerning arithmetic sequences in the video lesson that comes after that.

## 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.

A = 1 (the first term); d = 3 (the “common difference” across terms); A = 1 (the first term).

### 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 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.

Each arithmetic sequence is characterized by two coefficients that are unique to it: the common difference and the first term. 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.
- 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.
- (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.

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,.

If you want to get the n-th term of the arithmetico-geometric series, you must multiply the n-th term of the arithmetic progression by the n-th term of the geometric progression, which is the n-th term of the geometric progression. In this situation, the outcome will look somewhat like this:

- 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!

## 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

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! 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

- When the difference between successive terms is constant, this is referred to as an arithmetic sequence. In terms of its initial term, common differenced, and indexnas follows, the general term of an arithmetic sequence is stated as follows: an=a1+(n−1)d
- When you add up the terms of an arithmetic sequence, you get an arithmetic series. The partial sum of an arithmetic series may thus be computed using the first and last terms in the following manner: Sn=n(a1+an)2

- 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.

- 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)
- Find the sum of the first 200 positive integers in the sequence
- And The first 400 positive integers are added together to get a sum.

### 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

- 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 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

An arithmetic sequence is a list of integers in which each number is equal to the preceding number plus a constant, as defined by the definition above. The common difference ($d$) is a constant that is used to compare two things. Formulas:The $n$ term of an arithmetic progression may be found using the $color$ formula, where $color$ is the first term and $color$ is the common difference between the first and second terms. These are the formulae for calculating the sum of the first $n$ numbers: $colorleft(2a 1 + (n-1)d right)$ and $colorleft(a 1 + a right)$, respectively.

## Number Sequence Calculator

a n is defined as a 1 plus f. (n-1) For instance, the numbers 1, 3, 5, 7, 9, 11, and 13 are all numbers.

## Geometric Sequence Calculator

For example, the numbers 1, 2, 4, 8, 16, 32, 64, and 128 are all represented by the symbol a n.

## Fibonacci Sequence Calculator

The following are examples: 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Definition: A 0 equals zero; A 1 equals one; A n equals an n-1 plus an n-2. In mathematics, a sequence is a collection of things that are arranged in a specific order. As a result, a number sequence is defined as an ordered set of numbers that follow a specific pattern in some way. The individual pieces of a sequence are commonly referred to as terms, and the total number of terms in a sequence is referred to as its length, which can be infinite in some instances.

Innumerable distinct types of number sequences exist, with the most frequent being arithmetic sequences, geometric sequences, and Fibonacci sequences being three of the most prevalent forms.

An infinite series is called convergent if it converges to a certain limit, whereas a sequence that does not converge is known as divergent.

They are particularly helpful as a beginning point for series (which, in essence, explain the action of adding infinite quantities to a starting quantity), which are commonly employed in differential equations and the branch of mathematics known as analysis.

When dealing with patterns that are more complicated, indexing is typically the recommended method to use. When you index a sequence, you are constructing a generic formula that allows you to determine the then thterm of a series as a function of the number n.

### Arithmetic Sequence

When it comes to number sequences, an arithmetic series is one in which the difference between each consecutive term remains constant. Because of this disparity, the arithmetic series will either go towards positive or negative infinity, depending on the sign of the difference between the two values. The generic form of an arithmetic sequence is represented by the notation:

a n= a 1+ f × (n-1)or more generally | wherea nrefers to then thterm in the sequence | |

a n= a m+ f × (n-m) | a 1is the first term | |

i.e. | a 1, a 1+ f, a 1+ 2f,. | fis the common difference |

EX: | 1, 3, 5, 7, 9, 11, 13,. |

It is evident from the preceding sequence that the common differencef is number two. Using the equation above, we can compute the fifth term as follows:

EX: | a 5= a 1+ f × (n-1)a 5= 1 + 2 × (5-1)a 5= 1 + 8 = 9 |

When the sequence is compared to the equation, it can be observed that the 5th term, a 5, which was discovered using the equation, fits the sequence exactly as predicted. The sum of an arithmetic series may also be computed in a straightforward and straightforward manner using the following formula, in conjunction with the prior approach to finda n: Determine how many terms are in the arithmetic series through the 5 thterm, using the same numerical sequence as in the preceding example:

EX: | 1 + 3 + 5 + 7 + 9 = 25(5 × (1 + 9))/2 = 50/2 = 25 |

### Geometric Sequence

When a number series begins with a single number, it is called a geometric sequence. Each succeeding number following the initial number is the product of the preceding number multiplied by a fixed, non-zero integer (common ratio). The following is an example of the general form of a geometric sequence:

a n= a × r n-1 | wherea nrefers to then thterm in the sequence | |

i.e. | a, ar, ar 2, ar 3,. | ais the scale factor andris the common ratio |

EX: | 1, 2, 4, 8, 16, 32, 64, 128,. |

According to the preceding example, the common ratioris 2 and the scale factorais 1. Calculate the eighth term based on the equation provided above: When the value obtained from the equation is compared to the geometric sequence shown above, it is confirmed that they are identical. The following is the mathematical equation for computing the sum of a geometric sequence: Determine how much the total of the geometric sequence through the 3 rdterm is using the same geometric sequence as previously.

### Fibonacci Sequence

When the value obtained from the equation is compared to the geometric sequence described above, it is confirmed that they are identical. In order to calculate the sum of a geometric series, the following equation is used: Determine how much the total of the geometric sequence through the third term is using the same geometric sequence as above. For example, 1 plus 2 plus 4 equals 7.

a n= a n-1+ a n-2 | wherea nrefers to then thterm in the sequence | |

EX: | 0, 1, 1, 2, 3, 5, 8, 13, 21,. | a 0= 0; a 1= 1 |

## How to Find Any Term of an Arithmetic Sequence

Documentation Download Documentation Download Documentation An arithmetic sequence is a collection of integers that differ from one another by a fixed amount from one to the next. Consider the following example: the list of even integers. This is an arithmetic sequence since the difference between one number in the list and the next is always 2. It is possible to be requested to discover the very next phrase from a list of terms if you are aware that you are working with an arithmetic sequence.

You can also be asked to fill in a blank if a phrase has been left out. Finally, you could be interested in knowing, for example, the 100th phrase without having to write down all 100 words one by one. You may do any of these tasks with the aid of a few easy steps.

- 1 Determine the common difference between the two sequences. A list of numbers may be given to you with the explanation that the list is an arithmetic sequence, or you may be required to figure it out for yourself. In each scenario, the initial step is the same as it is in the other. Choose the first two terms that appear consecutively in the list. Subtract the first term from the second term to arrive at the answer. It is the outcome of your sequence that is the common difference
- 2 Check to see if the common difference is constant across the board. Finding the common difference between the first two terms does not imply that your list is an arithmetic sequence in the traditional sense. You must ensure that the difference is continuous across the whole list. Subtract two separate consecutive terms from the list to see how much of a difference there is. If the result is consistent for one or two other pairs of words, then you have most likely discovered an arithmetic sequence of terms. Promotional material
- 3 Add the common difference to the last phrase that was supplied. Finding the next term in an arithmetic series is straightforward after you’ve determined the common difference. Simply add the common difference to the final phrase in the list, and you will arrive at the next number in the sequence. Advertisement

- 1 Double-check that you are starting with an arithmetic sequence before proceeding. Sometimes you will have a list of numbers with a missing phrase in the center, and this will be the case. As with the last step, begin by ensuring that your list is an arithmetic sequence. Make a choice between any two consecutive words and calculate the difference between them. Once you’ve done that, compare it to two additional consecutive terms in the list. You can proceed if the differences are the same, in which case you can assume you are working with an arithmetic series. 2 Before the space, add the common difference to the end of the word. This is analogous to appending a phrase to the end of a sequence of words. Locate the phrase in your sequence that comes directly before the gap in question. This is the “last” number that you are familiar with. By multiplying this term by your common difference, you may get the number that should be used to fill in the blank
- 3 To calculate the common difference, subtract it from the phrase that follows the space. Check your response from the other way to be certain that you have the proper answer. An arithmetic sequence should be consistent in both directions, regardless of the direction in which it is performed. If you travel from left to right and add 4, then you would proceed in the opposite direction, from right to left, and do the reverse and remove 4
- 4 is the sum of the two numbers. You should compare your results. The two outcomes that you obtain, whether you add up from the bottom or subtract down from the top, should be identical to one another. If they do, you have discovered the value for the word that was previously unknown. It is your responsibility to ensure that your work is error-free. The arithmetic sequence you have may or may not be correct. Advertisement

- 1 Identify the first phrase in the series by looking at the initial letter of the word. Not all sequences begin with the integers 0 or 1 as the first or second numbers. Take a look at the list of numbers you have and identify the first phrase on it. Your beginning position, which can be identified using variables such as a(1), is the following: 2Define your common difference as d in the following way: Find the common difference between the sequences, just like you did previously. The common difference in this working example is 5, which is the most significant. It is the same result if you check with any of the other words in the sequence. This is a common distinction between the algebraic variable d, which we shall observe. 3 Use the explicit formula to solve the problem. In algebra, an explicit formula is a mathematical equation that may be used to determine any term in an arithmetic series without having to write down the entire list of terms in the sequence. For an algebraic series, the explicit formula is written as

- It is possible to read the word a(n) as “the nth term of a,” where n denotes the number in the list that you are looking for and a(n) reflects the actual value of that number. The number n will be 100 if you are asked to locate the 100th item in an arithmetic series, for example. Notably, while n is 100 in this example, the value of the 100th term, rather than the number 100 itself, will be represented as a(n).

4 Fill in the blanks with your information to help us solve the problem. Make use of the explicit formula for your sequence to enter the information that you already know in order to locate the word that you want. Advertisement

- 1, rearrange the explicit formula such that it may be used to solve for additional variables. Several bits of information about an arithmetic sequence may be discovered by employing the explicit formula and some fundamental algebraic operations. As written in its original form, the explicit formula is intended to solve for an integer n and provide you with the nth term in a series of numbers. You may, however, modify this formula algebraically and solve for any of the variables in the equation. 2 Find the first phrase in a series by using the search function. For example, you may know that the 50th term of an arithmetic series is 300, and you may also know that the terms have been growing by 7 (the “common difference”), but you may wish to know what the sequence’s very first term was. To determine your solution, use the new explicit formula that solves for a1. This formula is available online.

- Make use of the equation and fill in the blanks with the facts you already know. Because you know that the 50th term is 300, n=50, n-1=49, and a(n)=300 are the values of n. You are also informed that the common difference, denoted by the letter d, is seven. Therefore, the formula is as follows: This works out as well. The series that you have created began at 43 and increased by 7 each time. As a result, it appears as follows: 43,50,57,64,71,78.293,300

3 Determine the total length of a sequence. Consider the following scenario: you know the beginning and ending points of an arithmetic series, but you need to know how long it is. Make use of the updated formula.

- Consider the following scenario: you know that a specific arithmetic sequence starts at 100 and grows by 13. In addition, you are informed that the ultimate term is 2,856. You can find out the length of the series by putting the terms a1=100, d=13, and a(n)=2856 together. Fill in the blanks with the terms from the formula to get the answer. If you do the math, you will come up with, which equals 212+1, which equals 213. 213 words are included inside a single sequence
- An example of this would be the following: 101-313-126-213-136-139.2843-2856.

Create a new question

- Question How can I determine the first three terms if I only have the tenth and fifteenth terms? Subtract the tenth term from the fifteenth term and divide by five to get D, which is the difference between any two consecutive terms in the series of terms. Calculate the first term by multiplying D by 9 and subtracting that amount from the tenth term
- This is the first term. Question What is the mathematical formula for the numbers 8, 16, 32, 64, and ? This is not an arithmetic sequence in the traditional sense. Research geometric sequences for any formula you’re interested in learning about. Question How do I compute the 5 terms of an arithmetic sequence if the first term is 8 and the final term is 100, and the first term is 8 and the last term is 100? Take 8 away from 100 to get 92. 92 divided by 4 equals (because with five terms there will be four intervals between the first and last term). This gives you the number 23, which is the length of each interval. As a result, the sequence starts with 8 and has a common difference of 23
- Question How can I find out which term in the arithmetic sequence has the value of -38 in it? The common difference (d) is equal to 4 minus 7 = -3. The first term (a) equals 7. The given period (t) equals -38. (n-1)d = t + (a + (n-1)d, or, -38 = 7 + (n-1)-3, is the formula for time. As a result, n=16, which means that -38 is the sixteenth term
- Question The first three terms of 4n+3 are as follows: The first three terms, starting with n = 1, are 7, 11, and 15
- Question In the sequence 1/2, 1, 2, 4, 8, what is the formula for determining the nth term in the sequence? Alexandre Lima’s full name is Alexandre Lima. Community Answer This is a geometric progression in which each phrase is computed by multiplying the previous term by a predetermined constant before proceeding to the next. When using the example, the constant (q) is two since 2 * (1/2) = one, 2 * one = two, and 2 * two equals four. The formula is: a = a1 x q(n-1)
- For example, a = 1/2 x 2 in the example (n-1). For example, the tenth term is written as a(10) = 1/2 x 2(9) = 256. Question What is the best way to discover the 100th term if I only have the first five terms available? Take a look at Method 3 above, particularly Step 3. Question What if you have the common difference and the first term, but you need to know the a specific number is in relation to what nth number? For example, d=-4, a1=35, and 377 is a term number, correct? The formula for the nth term, denoted by the letter a(n), is provided in Method 3 above. Fill in the blanks with your numbers and solve for n
- Question What is the proper way to use the formula? If you want to discover the “nth” term in an arithmetic series, begin with the first term, which is a. (1). In addition, the product of “n-1” and “d” should be considered (the difference between any two consecutive terms). Consider the arithmetic sequences 3, 9, 15, 21, and 27 as an example. Because the difference between successive terms is always six, a(1) = three, and d = six. Consider the following scenario: you wish to locate the seventh word in the series (n = 7). Then a(7) = a(1) + (n-1)(d) = 3 + (6)(6) = 39, and a(7) = a(1) + (n-1)(d) = 39. In this sequence, number 39 corresponds to the seventh word
- Question What is the best way to locate the first three terms? Suppose you have the fourth, fifth, and sixth terms in the series, for example, 6, 8, and ten, respectively. The formula for finding any term in the series is Un (or Ur) = the first term + the term you are attempting to find minus one (for example, if you were trying to find the fifth term, the formula would be 5 -1) x d, where d is the length of the sequence (the common difference). Because you already know some of the terms in the sequence, you can put in the terms you already know into the formula and solve for the first term to get the answer: U(4) = 6 = U(1) + U(2) = U(4) (4-1) 2. The value of the fourth term, U(4), was provided as 6, and the common difference was found to be 2. After being simplified, the formula looks somewhat like this: 6 is equal to U(1) plus 6. The result of removing 6 from both sides is that U(1) equals 0, and you can use this to get any other term in the series using this formula.

More information on the replies Inquire about something There are 200 characters remaining. Include your email address so that you may be notified when this question has been resolved. SubmitAdvertisement

- There are several distinct types of number sequences to choose from. Do not make the mistake of assuming that a list of integers is an arithmetic series. Make sure to verify at least two pairings of words, and ideally three or four, in order to identify the common difference between them.

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- Remember that depending on whether it is being added or removed, the result might be either positive or negative.

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## About This Article

Summary of the Article When looking for a term in an arithmetic series, locate the common difference between the first and second numbers by subtracting the first from the second. Verify that the difference is consistent between each number in the series by re-running the preceding equation with the second and third numbers, third and fourth numbers, and so on until the difference is no longer consistent. Once you’ve determined the common difference, all that’s left to do to locate the missing number is to multiply the common difference by the term that came before it in the series.

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