What Is n in Arithmetic Sequence Formula? In the arithmetic sequence formula for finding the general term,an=a1+(n−1)d a n = a 1 + ( n − 1 ) d, n refers **to the number of terms in the given arithmetic sequence**.

Contents

- 1 What does N represent in sequences?
- 2 What does N represent in arithmetic sequence formula?
- 3 What is a sub n?
- 4 What is series and sequence?
- 5 How do you find the N term?
- 6 What is the sum of the first n terms of an arithmetic sequence?
- 7 Which of the following is the nth term of an arithmetic sequence?
- 8 What is the meaning of arithmetical?
- 9 How do you denote a subscript?
- 10 How do you read series and sequences?
- 11 What is the meaning of harmonic sequence?
- 12 Arithmetic Sequences and Sums
- 13 Arithmetic Sequence
- 14 Advanced Topic: Summing an Arithmetic Series
- 15 Footnote: Why Does the Formula Work?
- 16 Formulas for Arithmetic Sequences
- 17 Using Explicit Formulas for Arithmetic Sequences
- 17.1 A General Note: Explicit Formula for an Arithmetic Sequence
- 17.2 How To: Given the first several terms for an arithmetic sequence, write an explicit formula.
- 17.3 Example: Writing then th Term Explicit Formula for an Arithmetic Sequence
- 17.4 Try It
- 17.5 A General Note: Recursive Formula for an Arithmetic Sequence
- 17.6 How To: Given an arithmetic sequence, write its recursive formula.
- 17.7 Example: Writing a Recursive Formula for an Arithmetic Sequence
- 17.8 How To: Do we have to subtract the first term from the second term to find the common difference?
- 17.9 Try It

- 18 Find the Number of Terms in an Arithmetic Sequence
- 19 Solving Application Problems with Arithmetic Sequences
- 20 Contribute!
- 21 Arithmetic Sequences and Series
- 22 Arithmetic Sequences and Series
- 23 Arithmetic Series
- 24 Writing Rules for Arithmetic Sequences – Video & Lesson Transcript
- 25 Arithmetic Sequences
- 26 The Rule
- 27 Given Two Terms
- 28 Real World Applications of Arithmetic Sequences:
- 29 Arithmetic Sequences and Series – MathBitsNotebook(A2
- 30 Arithmetic sequence
- 31 Take the arithmetic sequence quiz below to check your understanding of this lesson.

## What does N represent in sequences?

Formulas give us instructions on how to find any term of a sequence. To remain general, formulas use n to represent any term number and a ( n ) a(n) a(n)a, left parenthesis, n, right parenthesis to represent the n th n^text{th} nthn, start superscript, start text, t, h, end text, end superscript term of the sequence.

## What does N represent in arithmetic sequence formula?

The values of a, d and n are: a = 1 (the first term) d = 3 (the “common difference” between terms) n = 10 ( how many terms to add up )

## What is a sub n?

A subscript is a character or string that is smaller than the preceding text and sits at or below the baseline. The small “_{n}” is a subscript. When used in the context “F_{n},” it refers to a function evaluated for the value “n.” The text _{n}_{–}_{1} and _{n}_{–}_{2} are also subscripts that define previous values of “n” in the sequence.

## What is series and sequence?

In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, or after, every other member. A series is a sum of a sequence of terms. That is, a series is a list of numbers with addition operations between them.

## How do you find the N 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.

## What is the sum of the first n terms of an arithmetic sequence?

The sum of the first n terms of an arithmetic sequence is called an arithmetic series.

## Which of the following is the nth term of an arithmetic sequence?

Answer: The expression to calculate the n^{th} term of an arithmetic sequence is a_{n} = a + (n – 1) d.

## What is the meaning of arithmetical?

(ærɪθmetɪkəl ) adjective [usually ADJECTIVE noun] Arithmetical calculations, processes, or skills involve the addition, subtraction, multiplication, or division of numbers.

## How do you denote a subscript?

Use “_” (underscore) for subscripts.

## How do you read series and sequences?

A sequence is defined as an arrangement of numbers in a particular order. On the other hand, a series is defined as the sum of the elements of a sequence.

## What is the meaning of harmonic sequence?

harmonic sequence, in mathematics, a sequence of numbers a_{1}, a_{2}, a_{3},… such that their reciprocals 1/a_{1}, 1/a_{2}, 1/a_{3},… The sum of a sequence is known as a series, and the harmonic series is an example of an infinite series that does not converge to any limit.

## Arithmetic Sequences and Sums

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

## Arithmetic Sequence

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

### Example:

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

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

### Example: (continued)

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

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

And this is what we get:

### Rule

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

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

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

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

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

## Advanced Topic: Summing an Arithmetic Series

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

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

The values ofa,dandnare as follows:

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

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

## Footnote: Why Does the Formula Work?

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

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

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

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

Each and every term is the same!

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

## Formulas for Arithmetic Sequences

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

## Using Explicit Formulas for Arithmetic Sequences

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

- Considering that the average difference is 50, the series represents a linear function with an associated slope of 50.
- You may also get the they-intercept by graphing the function and calculating the point at which a line connecting the points would intersect the vertical axis, as shown in the example.
- When working with sequences, we substitute _instead of y and ninstead of n.
- Using 50 as the slope and 250 as the vertical intercept, we arrive at this equation: = -50n plus 250 To create an explicit formula for an arithmetic series, we do not need to identify the vertical intercept of the sequence.

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

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

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

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

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

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

### Try It

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

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

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

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

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

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

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

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

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

No.

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

### Try It

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

## Find the Number of Terms in an Arithmetic Sequence

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

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

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

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

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

### Try It

The number of terms in the 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

In many application issues, it is typically preferable to begin with the term instead of_ as an initial condition. When solving these problems, we make a little modification to the explicit formula to account for the variation in beginning terms. The following is the formula that we employ: # # # # # # # # # # # # # # # # # # #

- 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 Series

HomeLessonsArithmetic Sequences and Series | Updated July 16th, 2020 |

Introduction |

Sequences of numbers that follow a pattern of adding a fixed number from one term to the next are called arithmetic sequences. The following sequences are arithmetic sequences:Sequence A:5, 8, 11, 14, 17,.Sequence B:26, 31, 36, 41, 46,.Sequence C:20, 18, 16, 14, 12,.Forsequence A, if we add 3 to the first number we will get the second number.This works for any pair of consecutive numbers.The second number plus 3 is the third number: 8 + 3 = 11, and so on.Forsequence B, if we add 5 to the first number we will get the second number.This also works for any pair of consecutive numbers.The third number plus 5 is the fourth number: 36 + 5 = 41, which will work throughout the entire sequence.Sequence Cis a little different because we need to add -2 to the first number to get the second number.This too works for any pair of consecutive numbers.The fourth number plus -2 is the fifth number: 14 + (-2) = 12.Because these sequences behave according to this simple rule of addiing a constant number to one term to get to another, they are called arithmetic sequences.So that we can examine these sequences to greater depth, we must know that the fixed numbers that bind each sequence together are called thecommon differences. Mathematicians use the letterdwhen referring to these difference for this type of sequence.Mathematicians also refer to generic sequences using the letteraalong with subscripts that correspond to the term numbers as follows:This means that if we refer to the fifth term of a certain sequence, we will label it a 5.a 17is the 17th term.This notation is necessary for calculating nth terms, or a n, of sequences.Thed -value can be calculated by subtracting any two consecutive terms in an arithmetic sequence.where n is any positive integer greater than 1.Remember, the letterdis used because this number is called thecommon difference.When we subtract any two adjacent numbers, the right number minus the left number should be the same for any two pairs of numbers in an arithmetic sequence. |

To determine any number within an arithmetic sequence, there are two formulas that can be utilized.Here is therecursive rule.The recursive rule means to find any number in the sequence, we must add the common difference to the previous number in this list.Let us say we were given this arithmetic sequence. |

Each arithmetic sequence has its own unique formula.The formula can be used to find any term we with to find, which makes it a valuable formula.To find these formulas, we will use theexplicit rule.Let us also look at the following examples.Example 1 : Let’s examinesequence Aso that we can find a formula to express its nth term.If we match each term with it’s corresponding term number, we get: |

The fixed number, which is referred to as the common differenceor d-value, is three. We may use this information to replace the explicit rule in the code. As an example, a n= a 1+ (n – 1)d. a n = a 1 + a (n – 1) the value of da n= 5 + (n-1) (3) the number 5 plus 3n – 3a the number 3n + 2a the number 3n + 2 When asked to identify the 37th term in this series, we would compute for a 37 in the manner shown below. the product of 3n and 2a 37 is 3(37) + 2a 37 is 111 + 2a 37 is 113. Exemple No. 2: For sequence B, find a formula that specifies the nth term in the series.

We can identify a few facts about it.Its first term, a 1, is 26.Itscommon differenceor d-value is 5.We can substitute this information into theexplicit rule.a n= a 1+ (n – 1)da n= 26 + (n – 1)(5)a n= 26 + 5n – 5a n= 5n + 21Now, we can use this formula to find its 14th term, like so. a n= 5n + 21a 14= 5(14) + 21a 14= 70 + 21a 14= 91ideo:Finding the nth Term of an Arithmetic Sequence uizmaster:Finding Formula for General TermIt may be necessary to calculate the number of terms in a certain arithmetic sequence. To do so, we would need to know two things.We would need to know a few terms so that we could calculate the common difference and ultimately the formula for the general term.We would also need to know the last number in the sequence.Once we know the formula for the general term of a sequence and the last term, the procedure involves the use of algebra.Use the two examples below to see how it is done.Example 1 : Find the number of terms in the sequence 5, 8, 11, 14, 17,., 47.This issequence A.In theprevious section, we found the formula to be a n= 3n + 2 for this sequence.We will use this along with the fact the last number, a n, is 47.We will plug this into the formula, like so.a n= 3n + 247 = 3n + 245 = 3n15 = nn = 15This means there are 15 numbers in this arithmetic sequence.Example 2 : Find the number of terms in the arithmetic sequence 20, 18, 16, 14, 12,.,-26.Our first task is to find the formula for this sequence given a 1= 20 and d = -2.We will substitute this information into theexplicit rule, like so.a n= a 1+ (n – 1)da n= 20 + (n – 1)(-2)a n= 20- 2n + 2a n= -2n + 22Now we can use this formula to find the number of terms in the sequence.Keep in mind, the last number in the sequence, a n, is -26.Substituting this into the formula gives us.a n= -2n + 22-26 = -2n + 22-48 = -2n24 = nn = 24This means there are 24 numbers in the arithmetic sequence. |

Given our generic arithmeticsequence.we can add the terms, called aseries, as follows.There exists a formula that can add such a finite list of these numbers.It requires three pieces of information.The formula is.where S nis the sum of the first n numbers, a 1is the first number in the sequence and a nis the nth number in the sequence.If you would like to see a derivation of this arithmetic series sum formula, watch this video.ideo:Arithmetic Series: Deriving the Sum FormulaUsually problems present themselves in either of two ways.Either the first number and the last number of the sequence are known or the first number in the sequence and the number of terms are known.The following two problems will explain how to find a sum of a finite series.Example 1 : Find the sum of the series 5 + 8 + 11 + 14 + 17 +. + 128.In order to use the sum formula.We need to know a few things.We need to know n, the number of terms in the series.We need to know a 1, the first number, and a n, the last number in the series.We do not know what the n-value is.This is where we must start.To find the n-value, let’s use the formula for the series.We already determined the formula for the sequence in a previous section.We found it to be a n= 3n + 2.We will substitute in the last number of the series and solve for the n-value.a n= 3n + 2128 = 3n + 2126 = 3n42 = nn = 42There are 42 numbers in the series.We also know the d = 3, a 1= 5, and a 42= 128.We can substitute these number into the sum formula, like so.S n= (1/2)n(a 1+ a n)S 42= (1/2)(42)(5 + 128)S 42= (21)(133)S 42= 2793This means the sum of the first 42 terms of the series is equal to 2793.Example 2 : Find the sum of the first 205 multiples of 7.First we have to figure out what our series looks like.We need to write multiples of seven and add them together, like this.7 + 14 + 21 + 28 +. +?To find the last number in the series, which we need for the sum formula, we have to develop a formula for the series.So, we will use theexplicit ruleor a n= a 1+ (n – 1)d.We can also see that d = 7 and the first number, a 1, is 7.a n= a 1+ (n – 1)da n= 7 + (n – 1)(7)a n= 7 + 7n – 7a n= 7nNow we can find the last term in the series.We can do this because we were told there are 205 numbers in the series.We can find the 205th term by using the formula.a n= 7na n= 7(205)a n= 1435This means the last number in the series is 1435.It means the series looks like this.7 + 14 + 21 + 28 +. + 1435To find the sum, we will substitute information into the sum formula. We will substitute a 1= 7, a 205= 1435, and n = 205.S n= (1/2)n(a 1+ a n)S 42= (1/2)(205)(7 + 1435)S 42= (1/2)(205)(1442)S 42= (1/2)(1442)(205)S 42= (721)(205)S 42= 147805This means the sum of the first 205 multiples of 7 is equal to 147,805. |

## 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

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

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

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

- Find a formula for the general term given 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=16anda15=76
- A4

### Part B: Arithmetic Series

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

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

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

### Part C: Discussion Board

- 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

## Writing Rules for Arithmetic Sequences – Video & Lesson Transcript

Yuanxin (Amy) Yang Alcocer is a Chinese actress. Amy, who holds a master’s degree in secondary education, has been working as a math teacher for more than nine years. Amy has experience working with kids of various ages and abilities, including those with special needs and those who are talented. Take a look at my bio Laura Pennington is a writer who lives in the United Kingdom. The University of Michigan awarded Laura a Master’s degree in Pure Mathematics after she earned a Bachelor’s degree in Mathematics from Grand Valley State University.

Take a look at my bio When creating arithmetic sequences in which terms proceed in a regular interval, there are a few general guidelines to remember to follow.

The most recent update was on December 15, 2021.

## Arithmetic Sequences

Yuanxin (Amy) Yang Alcocer is a Chinese-American actress and activist. Amy has a master’s degree in secondary education and has been working as a math teacher for more than nine years in various capacities. Amy has experience working with kids of various ages and abilities, from those with special needs to those who are very bright and creative. More information on the author Laura Pennington is a writer and actress who lives in the United Kingdom. The University of Michigan awarded Laura a Master’s degree in Pure Mathematics after she completed a Bachelor’s degree in Mathematics at Grand Valley State University.

More information on the author Arithmetic sequences in which terms proceed in a regular interval can be written according to a set of guidelines.

12/15/2021 (latest update) These sequences may not appear to be utilized for anything significant, yet they are in fact useful in a variety of situations and situations.

If the park charges an entrance fee of $10.00 as well as a per ride fee of $2.00, your total cost will be calculated as follows based on the number of rides you wish to ride: For no rides, the arithmetic sequence begins with $10, then proceeds on to $12 for one ride, then $14 for two rides, and so on.

## The Rule

For the reason that all arithmetic sequences follow the same pattern, you may apply a generic formula to obtain the formula for any particular sequence. The formula is as follows: Thea nrefers to the terms of the sequence, and thenrefers to the position of the term in the series.

This word refers to the first term in the sequence if nis is equal to 1. The difference between all of the consecutive integers in your series is represented by the letter d. The explicit formula for an arithmetic series is referred to as the explicit formula.

## Given Two Terms

Arithmetic sequences contain the same difference between succeeding pairs of terms in the sequence; as a result, you only need to know the first two terms of the series to construct the formula; the further terms of the sequence are not required. Let’s have a look at this. Take a look at this situation. Create a formula for the arithmetic series that begins with the numbers 4, 7, and so on. Only the first two words are provided to you. Because you already know the explicit formula rule, all you need to know is the first term and the difference between each succeeding pair of terms in the following formula.

You also know that the difference between the first term and the second term is 7.

- An=a1+d(n-1)
- An= 4 + 3 (n-1)
- An= 4 + 3 n- 3
- An= 3 n+ 1
- An= 4 n+ 1

You should leave the then s alone because they will always be a variable. Thesens are what allow you to utilize this formula to locate the remaining terms in your sequence using the information in this formula. They are the term’s location number in relation to the location number you are looking for. The rule for finding the formula for an arithmetic sequence reveals that your arithmetic sequence follows the explicit rule 3 n+ 1 for all of its terms, as demonstrated in the following example.

Consequently, in order to determine what the tenth term of the sequence is, all you need to do is punch in a 10 fornand evaluate:

## Real World Applications of Arithmetic Sequences:

- In an arithmetic sequence, the explicit formula for the then th term is defined asa n =a 1 + d(n – 1), wherea n is the then th term of the sequence, a 1 is the first term of the series, and dis the common difference of the sequence.

### Applications:

- Nancy is putting money aside to purchase a bike that will cost $275. She begins with $50 and continues to add $15 at the end of each week until she reaches her goal. It will take her about how many weeks to save up enough money to purchase the bike. Bob decided to start running as a New Year’s Resolution on January 1st, with the objective of running for one hour, or 60 minutes, straight. He begins by running for 5 minutes on the first day, and he increases his jogging time by 2 minutes on each subsequent day after that, until he reaches his goal. Bob’s objective is to attain it before the end of the month (which is 31 days from now).

### Answers:

- It will take 16 weeks to save $275 in this situation. For example, if she starts with $50 and adds $15 each week, the amount of money she has saved at the end of each week follows the mathematical sequence of 50 (first week), 65 (second week), 90 (third week), 105 (fourth week), and so on. In light of the fact that the bike costs $275, we are interested in knowing what term will be 275, or for what valuenwilla n = 275. The equation 275 = 50 + 15 is obtained by plugging these values into our explicit formula (n – 1). Solving the forngivesn=16 equation Yes. To demonstrate that this is the case, we check to see if Bob is still running for 60 minutes or more every day after 31 days of following this routine. He starts with 5 minutes and adds 2 minutes each day, therefore this may be represented mathematically by an arithmetic sequence with a beginning term of 5 and a common difference of 2 as shown in the diagram (or 5, 7, 9, 11,.). To determine if the 31st term will be more than or equal to 60, we pluga 1 = 5,d= 2, and n = 31 into our explicit formula to obtaina (31) = 5 + 2 (as in the explicit formula) (31 – 1). Bob can run for 65 minutes straight by the end of the month, which is more than 60 minutes, according to the simplified formula (31) = 65.

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## Arithmetic Sequences and Series – MathBitsNotebook(A2

Some sequences are composed of simply random values, while others have a definite pattern that is used to arrive at the sequence’s terms. Thearithmetic sequence(or progression),for example, is based upon the addition of a constant value to arrive at the next term in the sequence. |

When arithmetic sequences are graphed, they exhibit an alinear character (as a scatter plot). The domain of the sequence is represented by the counting numbers 1, 2, 3, 4, and so on (representing the position of each term), while the range of the sequence is represented by the actual terms of the series. While the x-axis rises by a constant value of one in the graph shown above, the y-axis increases by a constant value of three in the graph displayed above. Arithmetic Sequences are a type of mathematical sequence.

Arithmetic Sequence: | Common Difference,d: |

1, 6, 11, 16, 21, 26,. | d= 5.A 5 isaddedto each term to arrive at the next term.OR. thedifferencea2-a1= 5. |

10, 8, 6, 4, 2, 0, -2, -4,. | d= -2.A -2 isaddedto each term to arrive at the next term.OR. thedifferencea2-a1= -2. |

d= -½.A -½ isaddedto each term to arrive at the next term.OR. thedifferencea2-a1= -½. |

This is the sum of the firstnterms. |

S n=a1+(a1+ d)+(a1+2 d)+(a1+3 d)+(a1+4 d)+(a1+5 d)+(a1+6 d)+(a1+7 d)+(a1+8 d)+(a1+9 d)+(a1+(n- 1) d)+(a1+(n- 1) d)+(a When the terms of an arithmetic sequence are added together, the result is called an anarithmetic series.

Formulas that are used in conjunction with arithmetic sequences and arithmetic series include:

Tofind any term of anarithmetic sequence:wherea1is the first term of the sequence, dis the common difference,nis the number of the term to find. Note:you may seea1simply referred to asa. | To find thesum of a certain number of termsof anarithmetic sequence:whereSnis the sum ofnterms (nthpartial sum),a1is the first term,a nis thenthterm. Note:(a1+a n)/2 is the mean (average) of the first and last terms. The sum can be thought of as thenumber of terms times the average of the first and last terms. This formula may also appear as |

+(a1+(n- 1) d) a1+(a1+(n- 1) d) a1+(a1+2 d) +(a1+3 d) a1+4 d) + (a1+(n- 1)d) + (a1+(n- 1) d) S n=a1+(a1+ d) a1+(a1+2 d) a1+3 (a1 The addition of the terms of an arithmetic sequence is known as an arithmetic series. Calculations involving arithmetic sequences and arithmetic series include the following.

Questions: | Answers: |

1.Find the common difference for this arithmetic sequence: 4, 15, 26, 37,. | The common difference,d,can be found bysubtracting the first term from the second term, which in this example yields 11. Checking shows that 11 is the difference between all of the terms. |

2.Find the common difference for the arithmetic sequence whose formula is:a n= 6 n+ 3. | A listing of the terms will show what is happening in the sequence (start withn= 1). 9, 15, 21, 26, 33,.The common difference is 6. |

3.Find the 10 thterm of the sequence:3, 5, 7, 9, 11,. | By observation_n= 10,a1= 3,d= 2Use the formula for thenthterm.The 10 thterm is 21. |

4.Finda7for an arithmetic sequence where_a1= 3 xandd = -x. | By observation_n= 7,a1= 3 x,d= – xYour answer will be in terms of x. |

5.Given the arithmetic sequence:f(1) = 4;f(n) =f(n- 1) + 3.Findf(5). | Don’t let the change tofunctional notationdistract you. This problem showsrecursive form:each term is defined by the term immediately in front of it.The first term is 4 and the common difference is 3. Since we only need the fifth term, we can get the answer by observation: 4, 7, 10, 13, 16f(5) = 16 |

6.Findt15for an arithmetic sequence where_t3= -4 + 5i andt6= -13 + 11 iNOTE:Using high subscript – low subscript + 1 will count the number of terms. | Notice the change of labeling fromatot.The letter used in labeling is of no importance. Let’s get a visual of this problem. Using the third terms as the “first” term, find the common difference from these known terms.Now, fromt3tot15is 13 terms. t15= -4 + 5 i+ (13-1)(-3 +2 i) = -4 + 5 i-36 +24 i = -40 + 29 i |

7.Find an explicit formula and a recursive formula for the sequence:1, 3, 5, 7, 9,. | Theexplicit formulaneeds to relate the subscript number of each term to the actual value of the term. These terms are odd numbers (a good formula pattern to remember).a n= 2 n- 1Substitutingn= 1 gives 1.Substitutionn =2 gives 3, and so on.Therecursive formula,where each term is based upon the term immediately in front of it, is easy to find since the common difference is 2.a1= 1a n= an -1+ 2. |

8.The first three terms of an arithmetic sequence are represented byx+ 5, 3 x+ 2, and 4 x+ 3 respectively. Find the numerical value of the 10 thterm of this sequence. | Represent the common difference between the terms:(3 x+ 2) – (x+ 5) = 2 x- 3 (the common difference)(4 x+ 3) – (3 x+ 2) =x+ 1 (the common difference)Since the common difference must be constant, weset these values equal and solve forx.2 x- 3 =x+ 1 x= 4 The sequence is 9, 14, 19,., common difference of 5.The 10 thterm = 9 + (10 – 1)(5) =54 |

9.Find the sum of the first 12 positive even integers.Notice how BOTH formulas work together to arriveat the answer. | The word “sum” indicates the need for the sum formula.positive even integers: 2, 4, 6, 8,. n= 12,a1= 2,d= 2We are missinga12, for the sum formula, so we use the “any term” formula to find it.Now, we use this information to find the sum: |

10.Insert 3 arithmetic means between 7 and 23.Note:In this context, anarithmetic meanis the term between any two terms of an arithmetic sequence. It is simply the average (mean) of its surrounding terms. | While there are several solution methods, we will use our arithmetic sequence formulas. Draw a picture to better understand the situation.7, _, _, _, 23This set of terms is an arithmetic sequence.We know the first term,a1, the last term,a n, but not the common difference,d.This question gives NO indicationof “sum”, so avoid that formula. Find the common difference:Now, insert the terms usingd: 7,11, 15, 18, 23 |

11.In an arithmetic sequence,a4= 19 and a7= 31. Determine the formula fora n, thenthterm of this sequence. | Visualize the problem by modeling the terms from the fourth to the seventh.19, _. _, 31Temporarily imagine that 19 is the first term.This will allow us to find the common difference.Imagined Observations_a1= 19,a 4= 31,n= 4. |

12.Find the number of terms in the sequence:7, 10, 13,., 55Note:nmustbe an integer! | By observation_a1= 7,a n= 55,d= 3. We need tofindn. This question makes No mention of “sum”, so avoid that formula.When solving forn,be sure your answer is a positiveinteger.There is no such thing as a fractional number of terms in a sequence! |

13.A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater? | The seating pattern is forming an arithmetic sequence:60, 68, 76,.We need to find the “sum” of all of the seats. By observation_n= 20,a1= 60,d= 8 and we needa20for the sum.Now, use the sum formula.There are 2720 seats. |

## Arithmetic sequence

Before discussing arithmetic sequence, it is important to understand that in mathematics, a sequence is a collection of numbers that follow a pattern. Each number in the series is referred to as a word. As an illustration, consider the following sequences: 1, 4, 7, 10, 13, 16, and 19 are the first four digits of the number 1. 70, 62, 54, 48, 40, and so forth. When you have an arithmetic sequence, you can find each term by adding or subtracting the same value from one term to the next, and so on.

Taking a close look at the numbers 70, 62, 54, 46, and 38, for example, allows us to make the following observation: This time, to locate each word, we remove 8, which is a common difference between the previous term and the current term.

Here’s the trick, or rather, the recipe, as it were!

Allow any term number in the series to be represented by n.

The number -2 is the one that occurs immediately before 1 in the sequence.

Whereas n = 2, which represents the second term, we obtain 3 2 + -2 = 6 + -2 = 4 when n = 2 represents the first term.

Allow any term number in the series to be represented by n.

The number 78 appears immediately before the number 70 in the sequence.

When n = 1, which represents the first term, we obtain When n = 2, which represents the second term, we get -8 2 + 78 = -16 + 78 = 62.