sequence determined by a = 2 and d = 3. 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 What is the formula for sequence?
- 2 What is the formula for the sum of an arithmetic sequence?
- 3 What is example of arithmetic sequence?
- 4 How do you find the 25th term of an arithmetic sequence?
- 5 How do you find the arithmetic mean?
- 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 calculator that shows work
- 21 About this calculator
- 22 Arithmetic Sequence Calculator
- 23 What is an arithmetic sequence?
- 24 Arithmetic sequence definition and naming
- 25 Arithmetic sequence examples
- 26 Arithmetic sequence formula
- 27 Difference between sequence and series
- 28 Arithmetic series to infinity
- 29 Arithmetic and geometric sequences
- 30 Arithmetico–geometric sequence
- 31 Arithmetic sequence calculator: an example of use
- 32 Arithmetic Sequences Problems with Solutions
- 33 Review of Arithmetic Sequences
- 34 Problems with Solutions
- 35 Exercises
- 36 Solutions to Above Exercises
- 37 More References and links
- 38 Arithmetic Sequences
- 39 13.2: Arithmetic Sequences
- 40 Finding the Number of Terms in a Finite Arithmetic Sequence
- 41 Solving Application Problems with Arithmetic Sequences
- 42 How to Solve an Arithmetic Sequence Problem With Variable Terms
- 43 How to Derive the Arithmetic Sequence Formula
- 44 A Sample Arithmetic Sequence Problem
- 45 Arithmetic Sequences and Series
- 46 Arithmetic Series

## What is the formula for sequence?

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 for the sum of an arithmetic sequence?

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

## 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.

## 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.

## How do you find the arithmetic mean?

One method is to calculate the arithmetic mean. To do this, add up all the values and divide the sum by the number of values. For example, if there are a set of “n” numbers, add the numbers together for example: a + b + c + d and so on. Then divide the sum by “n”.

## 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

For the following arithmetic sequence, provide an explicit formula for it. left An arecursive formula is used to define some arithmetic sequences in terms of the previous phrase. When finding the terms of the sequence, the formula gives an algebraic procedure that may be applied. A recursive formula allows us to locate any term in an arithmetic series by utilizing a function of the term that came before it in the sequence. It is calculated by adding up each term’s previous term and the common difference between them.

In this case, if the common difference is 5, then each word is equal to the preceding term + 5. Whenever a recursive formula is used, it is necessary to provide the initial term. Beginning with the letter _, and ending with the letter _, we have the expression

### 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

A sequence is a collection of items (typically numbers) that are arranged in a certain order. ReadSequences and Seriesfor further information on what each number in the sequence is referred to as aterm (or “element” or “member”).

### 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

There are no variations! As well as the fact that there are “n” of them The number 2S equals the sum of the numbers (2a + (n1)d) and the number 2S equals two times the number two. Simply divide the result by two to obtain: The function S = (n/2) (2a + (n1)d) is defined as follows: What we’ve come up with is this:

## 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?

Determine whether or not the sequence is an AP, and then perform the simple procedures outlined below, which vary based on the values known or provided:

- 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 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.

## 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,.

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 Problems with Solutions

Arithmetic sequences are employed across mathematics and are applied to issues in engineering, physics, computer science, biology, and finance, among other fields. We give a series of questions and exercises utilizing arithmetic sequences, as well as thorough answers to the difficulties.

## Review of Arithmetic Sequences

An arithmetic sequence with one common differenced term and one initial term is represented by the expression [a n = (n – 1) d] and the formula for the next term is represented by arithmetic sequences are defined by the formula [s n = a 1 + a 2 + a 3 +.

+ a n] and their sums are obtained by the formula [s n = dfrac ] respectively. Calculate the Arithmetic Series Online Using This Calculator. Using an online calculator, you may figure out the total of the terms in an arithmetic series.

## Problems with Solutions

Problem 1: The first term of an arithmetic series is equal to 6 and the common difference is equal to 3. What is the common difference? Formula for the nth term and the value of the 50th term must be discovered. Solution to Problem 1: In the formula for the n th term provided above (a n = a 1 + (n – 1) d = 6 + 3 (n – 1) d = 3 n + 3 ), substitute the value of the common difference d = 3 and the first term a 1= 6 for the value of the common difference d = 3. The 50th term is determined by changing the value of n to 50 in the preceding formula.

- Calculate the value of the twentieth phrase.
- 20 = 200 minus (-10) (20 minus 1) Equals 10.
- Find out what the 15th term is.
- Knowing the first term and its common difference, we may apply the n th term formula to obtain the 15 th term, as shown in the following example.
- Find the 100th phrase in the sentence.
- a the sum of a 1+ (15 – 1) d = 62 We obtain a system of two linear equations in which the unknowns are denoted by the letters a and d.
- Find the value of d.d = 4 by solving the equation Then, using the value of d in one of the equations, calculate the value of a 1.a 1+ (5 – 1) 4 = 22.
- 6 + 4 (100 – 1) Equals 402 for the number a 100.
- Solution to Problem 5: The series of integers beginning with 1 and ending with 1000 is represented by the numbers 1, 2, 3, 4,., 1000.
- Because the first word is one and the last term is a thousand, the common difference is exactly one.

s 1000= 1000 (1 + 1000) / 2 = 500500 s 1000= 1000 (1 + 1000) / 2 = 500500 s 1000= 1000 (1 + 1000) / 2 = 500500 s 1000= 1000 (1 + 1000) / 2 = 500500 s 1000= 1000 (1 + 1000) / 2 = 500500 s 1000= 1000 (1 + 1000) / 2 = 500500 s 1000= 1000 (1 + 1000) / 2 = 500500 s 1000= 1000 (1 + 1000 Problem 6: Calculate the sum of the first 50 even positive numbers in the given range.

- The preceding sequence has a first term with a value of 2 and a common difference with a value of 2.
- The sum of two plus two (50 minus one) equals one hundred.
- 50= 50 (2 + 100) / 2 = 2550.
- Calculate the sum of all positive integers that are divisible by 5, ranging from 5 to 1555 inclusive, that are divisible by 5.
- It is worth noting that the preceding sequence contains a first term equal to 5 and a common difference d = 5.
- In order to calculate the nth term, we utilize the following formula: 1555 is equal to a 1+ (n – 1)d Make substitutions for 1 and d using their respective values.
- Because we now know that 1555 is the 311th term, we can apply the following formula to get the total: The number 311 equals 311 (5 + 1555) / 2 = 242580.
- (2) (n = 2 sum_ n + sum_ n (half of 2)) The word n is made up of the sum of the first ten positive integers in the sequence.

It is possible to calculate this total by applying the formula s N= n (a 1+a n) / 2, which is written as 10(1+10)/2 = 55 is the answer. It is given by 10(1/2) = 5 that the term (1 / 2) is the result of the addition of a constant term 10 times. The total S may be calculated as S = 2(55) + 5 = 115.

## Exercises

Answer the following questions on arithmetic sequences in a concise manner: a) Determine the value of a 20 given that a 3 equals 9 and an 8 equals 24. b) Determine the value of 30 given that the first few terms of an arithmetic series are supplied by the numbers 6,12,18, and. C) Determine d provided that a 1= 10 and a 20= 466d) Determine s 30 given that a 10= 28 and a 20= 58e) Determine the sum S defined by [S = _sum_ (3n / 2)]f) Determine the sum S defined by f) Find the total S defined by [S = sum_ 0.2 n + sum_ 0.4 j] and the sum S defined by [S = sum_ 0.2 n + sum_ 0.4 j]

## Solutions to Above Exercises

A) a 20= 60b) a 30= 180c) d = 24d) s 30= 1335e) 1380f) 286 A) a 20= 60b) a 30= 180c) d = 24d) s 30= 1335e) 1380f) 286

## More References and links

- The following are some examples of math problems and their thorough answers: geometric sequences problems with solutions
- Math tutorials and questions
- Math problems with explicit solutions

## Arithmetic Sequences

This lesson will work with arithmetic sequences, their recursive and explicit formulas and finding terms in a sequence. In this lesson, it is assumed that you know what an arithmetic sequence is and can find a common difference. If you need to review these topics,click here.Let’s look at the arithmetic sequence20, 24, 28, 32, 36,.This arithmetic sequence has a common difference of 4, meaning that we add 4 to a term in order to get the next term in the sequence.The recursive formula for an arithmetic sequence is written in the formFor our particular sequence, since the common difference (d) is 4, we would writeSo once you know the common difference in an arithmetic sequence you can write the recursive form for that sequence.However, the recursive formula can become difficult to work with if we want to find the 50 thterm. Using the recursive formula, we would have to know the first 49 terms in order to find the 50 th. This sounds like a lot of work. There must be an easier way. And there is!Rather than write a recursive formula, we can write an explicit formula. Theexplicit formulais also sometimes called the closed form. To write the explicit or closed form of an arithmetic sequence, we usea nis the nth term of the sequence. When writing the general expression for an arithmetic sequence, you will not actually find a value for this. It will be part of your formula much in the same way x’s and y’s are part of algebraic equations.a1is the first term in the sequence. To find the explicit formula, you will need to be given (or use computations to find out) the first term and use that value in the formula.nis treated like the variable in a sequence. For example, when writing the general explicit formula, n is the variable and does not take on a value. But if you want to find the 12th term, then n does take on a value and it would be 12.dis the common difference for the arithmetic sequence. You will either be given this value or be given enough information to compute it. You must substitute a value for d into the formula.You must also simplify your formula as much as possible.Let’s Practice: |

- Write out the explicit formula for the sequence that we were working with earlier in the session.

20, 24, 28, 32, 36, and so on The first term in the series is 20 and the common difference between the two terms is four. This is sufficient information for constructing the explicit formula. In order to arrive to our final result, we must first simplify this statement. Consequently, the explicit (or closed) formula for the arithmetic sequence is as follows: It is important to note that the an nthe and n phrases did not take on numerical values. It’s the same as when x and y are used in algebraic expressions; they are a component of the formula once again.

Take a look at the following example to see what occurs.

- The sequence 20, 24, 28, 32, 36, and so on is used to discover the 50th phrase.

Find the 50th term in the following sequence: 20, 24, 28, 32, 36,.

- Find the explicit formula for a sequence in which d = 3 and a 12 = 58 is repeated three times.

As stated in the formula, we must be aware of the first term as well as the common difference. We have d, however we are unable to identify a 1. We do, however, have enough information to track it down. We already know that when n = 12, the 12th term in the series is 58, so we don’t have to worry about it. If we simplify the equation, we can come up with a number one. With the first term and the d value provided in this problem in hand, we may work our way through the issue to get the explicit formula.

While working backwards from an answer the first time, we used the formula to come up with an explicit formula the second time.

- The explicit formula for an arithmetic sequence with a 1 equaling 4 and a 2 equaling 10 must be discovered.

In this instance, we have the first term, but we are unable to determine the common difference. We do, however, know two consecutive terms, which implies we may get the common difference by subtracting the two terms together. The fact that 10 – 4 = 6 indicates that d = 6. Now we’ll utilize the formula to figure out how to acquire It is important to note that knowing the first term and the common difference before constructing an explicit formula is a given. If none of these are provided in the problem, you must use the information provided to track them down on your own.

The way to solve this problem is to find the explicit formula and then see if 623 is a solution to that formula.We already found the explicit formula in the previous example to be. To find out if 623 is a term in the sequence, substitute that value in for a n.What does this mean? Well, if 623 is a term in the sequence, when we solve the equation, we will get a whole number value for n. Since we did not get a whole number value, then 623 is not a term in the sequence. Look at it this way. There can be a 103 rdterm or a 104 thterm, but not one in between.Find the recursive formula for 15, 12, 9, 6,. | ||

What is your answer? | ||

Find the explicit formula for 15, 12, 9, 6,. | ||

What is your answer? | ||

Find the recursive formula for 5, 9, 13, 17, 21,. | ||

What is your answer? | ||

Find the explicit formula for 5, 9, 13, 17, 21,. | ||

What is your answer? | ||

Find a 10, a 35and a 82for problem4. | ||

What is your answer? | ||

Find the explicit formula whenand d = 2. | ||

What is your answer? | ||

Find a 10, a 35and a 82for problem6. | ||

What is your answer? | ||

Find the explicit formula whenand. | ||

What is your answer? | ||

Find a 10, a 35and a 82for problem8. | ||

What is your answer? | ||

Is 327 a term in the sequence 8, 13, 18,.? | ||

What is your answer? | ||

Is 852 a term in the sequence 8, 12, 16, 20,.? | ||

What is your answer? | ||

## 13.2: Arithmetic Sequences

Example (PageIndex): After writing the first Term, write the second Term. An Arithmetic Sequence with a Clearly Defined Formula Create an explicit formula for the arithmetic series using the following syntax: ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( It is possible to calculate the common difference by subtracting the first term from the second term. The most noticeable change is referred to as (10). To simplify the formula, substitute the common difference and the first term in the series into it.

Drawing (figure) (PageIndex ) Take part in an exercise program (PageIndex ) For the arithmetic series that follows, provide an explicit formula for it. Answer(a n=533n) (a n=533n) (a n=533n) (a n=533n)

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

When determining the number of terms in a finite arithmetic sequence, explicit formulas can be employed to make the determination. Finding the common difference and determining the number of times the common difference must be added to the first term in order to produce the last term of the sequence are both necessary steps. Calculate the total number of terms in a finite arithmetic sequence using the first three terms and the last term as inputs.

- Figure out what the common difference (d) is
- Replace the common difference and the first term in (a n=a 1+d(n–1)) with the common difference and the first term. Make a substitution for the final word in (a n) and solve for (n)
- A.

Figure 1: Finding the Number of Terms in a Finite Arithmetic Sequence using the PageIndex method. The number of terms in the finite arithmetic sequence has to be determined. ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( It is possible to calculate the common difference by subtracting the first term from the second term. (1−8=−7) The most often encountered difference is (7). Substitute the common difference and the first term of the series into the nth term formula, and then simplify the resultant formula.

There are a total of eight terms in the series.

answError: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: (Answer: There are a total of eleven terms in the series.

## Solving Application Problems with Arithmetic Sequences

In many application situations, it is typically preferable to utilize an initial term of (a 0) rather than (a 1) as the first term. In order to account for the variation in beginning terms in both cases, we make a little modification to the explicit formula. The following is the formula that we use: The following is an example of (PageIndex ): Problem-Solving using Arithmetic Sequences in Practical Situations Every week, a five-year-old child is given a monetary allowance of one dollar. His parents offer him a yearly raise of ($2 per week) on top of his current salary.

- Create a method for calculating the child’s weekly stipend over the course of a year
- What will the child’s allowance be when he reaches the age of (16) years?

- In this case, an arithmetic sequence with an initial term of (1,1) and a common difference of (1,1) may be used to simulate the scenario (2). Let (A) be the amount of the allowance and n denote the number of years after the age of retirement (5). This is what we get if we use the changed explicit formula for an arithmetic sequence: (A r=1+2n)
- By subtracting, we may get the number of years that have passed since the age of (5). (16−5=11) It is our intention to obtain the child’s allowance after eleven years. In order to calculate the child’s allowance at age, substitute (11) into the calculation (16). (A_ =1+2(11)=23) is a prime number. The child’s allowance will be ($23) per week when he or she reaches the age of sixteen.

Take part in an exercise program (PageIndex ) The next week, a lady chooses to go for a ten-minute run every day, with the goal of increasing the length of her daily exercise by four minutes each week. Formulate the time she will run after (n) weeks to determine the distance she will cover. How long will her daily run last in a year and a half from now? Answer The formula is (T n=10+4n,) and it will take her a total of (42) minutes to complete. In addition to further teaching and practice with arithmetic sequences, you can access this online resource for that purpose.

recursive formula for nth term of an arithmetic sequence | (a_n=a_ +d) (n≥2) |

explicit formula for nth term of an arithmetic sequence | (a_n=a_1+d(n−1)) |

- When there is a constant difference between any two consecutive terms in an arithmetic sequence, the sequence is called an arithmetic sequence
- The constant difference between two consecutive terms is known as the common difference
- The common difference is the number that is added to any one term of an arithmetic sequence in order to generate the subsequent term. See the following example: (PageIndex)
- The terms of an arithmetic sequence can be obtained by starting with the first term and adding the common difference over and over until the sequence is complete. See Examples ((PageIndex ) and ((PageIndex ) for more information. The recursive formula for an arithmetic series with common difference dd is provided by (a n=a_ +d), and (n2 is the number of steps in the sequence). See the following example: (PageIndex)
- As with any recursive formula, the first term in the series must be specified
- Otherwise, the formula will fail. It is possible to express an explicit formula for an arithmetic series with a common difference d using the formula (a n=a 1+d(n)1). See the following example: (PageIndex)
- When determining the number of words in a sequence, it is possible to apply an explicit formula. Observe the following example: (PageIndex)
- In application situations, we may slightly modify the explicit formula to (a n=a 0+dn). See the following example: (PageIndex)

## How to Solve an Arithmetic Sequence Problem With Variable Terms

Mathematical sequences may be defined as any collection of numbers that are organized in an orderly fashion. As an illustration, the numbers 3, 6, 9, 12,. Another example would be the numbers 1, 3, 9, 27, 81, and so on. The three dots on the right indicate that the set is not over. Each of the numbers in the set is referred to as a word. Each phrase in an arithmetic sequence is separated from the one before it by a constant, which is added to each term in the sequence. To obtain the following term in the first example, the constant is 3, and you add 3 to each of the terms.

Even when an arithmetic sequence is just a few terms long, figuring it out can be a simple task.

But what if the series is thousands of terms long and you need to discover a term somewhere in the middle? However, there is a far more convenient way to type out the sequence: using an electronic keyboard. You utilize the arithmetic sequence formula to do this.

## How to Derive the Arithmetic Sequence Formula

If you designate the initial term in an arithmetic series by the letter a and the common difference between terms is denoted by the letter d, you may express the sequence in the following format:a, (a + d), (a + 2d), (a + 3d),.a, (a + 2d), (a + 3d),. Using the denotation x n for the nth term in the series will allow you to create a generic formula for it, which is as follows: x n = a + d is a mathematical formula (n – 1) This may be used to discover the tenth term in the series of numbers 3, 6, 9, 12,.

## A Sample Arithmetic Sequence Problem

There are a variety of tasks in which you are supplied with a sequence of numbers, and you must utilize the arithmetic sequence formula to build a rule in order to derive any term from that particular sequence. For example, construct a rule for the numbers 7, 12, 17, 22, 27,. in the sequence 7. In this case, the common difference (d ) is 5 and the first term (a ) is 7. Simply plugging in the numbers and simplifying is all that is required to find the n th term, which can be found by using the arithmetic sequence formula: a + d = beginx n a + d = beginx n (n – 1) = 7 + 5 = 14 (n – 1) the sum of the squares of the numbers 7 and 5n minus five is equal to the sum of the squares of the numbers 2 and 5n.

Consider the following scenario: If you’re looking for the 100th term ( x 100), then n = 100 and the term in question is 502.

The number 377 corresponds to the 75th word in the series.

## 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

4. Determine whether or not there is a common difference between the first 50 terms of the provided sequence: 4, 9, 14, 19, 24,.Solution: Determine whether or not there is a common difference between the first 50 terms of the given sequence: 4, 9, 14, 19, 24,. It is true that the sequence is an arithmetic progression, and we can writean=a1+(n1)d=4+(n1)5=4+5n5=5n1 As a result, the general term isan=5n1. To calculate the 50 thpartial sum of this sequence, we need to know the 1 stand for the 50 thterms_a1=4a50=5(50)1=249 Next, use the formula to determine the 50

### Example 6

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

### Example 7

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

where a1=26 and d=2.

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

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

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

### Key Takeaways

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

### Topic Exercises

- Given the first term and common difference of an arithmetic series, write the first five terms of the sequence. Find a formula that describes the generic term. The values of a1 are 5 and 3
- 12 and 2
- 15 and 5
- 7 and 4 respectively
- 12 and 1
- A1=23 and 13 respectively
- 1 and 12 respectively
- A1=54 and 14. The values of a1 are 1.8 and 0.6
- 4.3 and 2.1
- And a1=5.4 and 2.1 respectively.

- Locate a formula for the general term and apply it to get the 100 thterm, given the arithmetic series given the sequence 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.

- Find a formula for the general term from the terms of an arithmetic sequence given the terms of the series. 1 = 6 and 7 = 42
- 1 = 12 and 12= 6
- 1 = 19 and 26 = 56
- 1 = 9 and 31 = 141
- 1 = 16 and 10 = 376
- 1 = 54 and 11 = 654. 1 = 6 and 7 = 42
- 1= 9 and 31 = 141
- 1 = 6 and 7

- Find all of the arithmetic means that exist between the two supplied terms. a1=3anda6=17
- A1=5anda5=7
- A2=4anda8=7
- A5=12anda9=72
- A5=15anda7=21
- A6=4anda11=1

### Part B: Arithmetic Series

- In light of the general term’s formula, figure out how much the suggested total is. 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

- Evaluate. 1160(3n)
- 1121(2n)
- 1250(4n-3)
- 1120(2n+12)
- 170 (198n)
- 1220(5n)
- 160(5212n)
- 151(38+14
- 1120(1.5n+2.6)
- 1175(0.2N1.6)
- 1170 (19 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 numbers
- The sum of the first 200 positive odd integers
- The sum of the first 500 positive odd integers
- The sum of the first 50 positive even numbers
- The sum of the first 200 positive even integers
- The sum of the first 500 positive even integers
- The sum of the firstk positive odd integers
- The sum of the firstk positive odd integers the sum of the firstk positive even integers
- The sum of the firstk positive odd integers
- There are eight seats in the front row of a tiny theater, which is the standard configuration. Following that, each row contains three additional seats than the one before it. How many total seats are there in the theater if there are 12 rows of seats? In an outdoor amphitheater, the first row of seating comprises 42 seats, the second row contains 44 seats, the third row contains 46 seats, and so on and so forth. When there are 22 rows, how many people can fit in the theater’s entire seating capacity? The number of bricks in a triangle stack are as follows: 37 bricks on the bottom row, 34 bricks on the second row and so on, ending with one brick on the top row. What is the total number of bricks in the stack
- Each succeeding row of a triangle stack of bricks contains one fewer brick, until there is just one brick remaining on the top of the stack. Given a total of 210 bricks in the stack, how many rows does the stack have? A wage contract with a 10-year term pays $65,000 in the first year, with a $3,200 raise for each consecutive year after. Calculate the entire salary obligation over a ten-year period (see Figure 1). In accordance with the hour, a clock tower knocks its bell a specified number of times. The clock strikes once at one o’clock, twice at two o’clock, and so on until twelve o’clock. A day’s worth of time is represented by the number of times the clock tower’s bell rings.

### Part C: Discussion Board

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

### Answers

- An=3n+2
- An=5n+3
- An=6n
- An=3n+2
- An=6n+3
- An=6n+2

- 1,565,450, 2,500,450, k2,
- 90,800, k4,230,
- 38640, 124,750,
- 18,550, k765
- 10,578
- 20,100,
- 2,500,550, k2,
- 294 seats, 247 bricks, $794,000, and so on.