In arithmetic sequences, the common difference is **simply the value that is added to each term to produce the next term of the sequence.**

Contents

- 1 How do you find the common difference in arithmetic sequences?
- 2 What is the common difference arithmetic?
- 3 What is the example of common difference?
- 4 Whats a common difference?
- 5 What is the common difference in the sequence 3?
- 6 What is the common difference of the arithmetic sequence 3 13 4?
- 7 Common Difference – Formula, How to Find Common Difference?
- 8 What Is Common Difference?
- 9 Common Difference Formula
- 10 Finding Common Difference in Arithmetic Progression
- 11 Examples on Common Difference
- 12 FAQs on Common Difference
- 13 Finding Common Differences
- 14 Writing Terms of Arithmetic Sequences
- 14.1 How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.
- 14.2 Example 2: Writing Terms of Arithmetic Sequences
- 14.3 Solution
- 14.4 Analysis of the Solution
- 14.5 Try It 3
- 14.6 How To: Given any the first term and any other term in an arithmetic sequence, find a given term.
- 14.7 Example 3: Writing Terms of Arithmetic Sequences
- 14.8 Solution
- 14.9 Analysis of the Solution
- 14.10 Try It 4

- 15 Introduction to Arithmetic Progressions
- 16 Finite Sequence: Definition & Examples – Video & Lesson Transcript
- 17 Examples
- 18 Finding Patterns
- 19 Finding Sums of Finite Arithmetic Series – Sequences and Series (Algebra 2)
- 20 Common Difference: Formula & Overview – Video & Lesson Transcript
- 21 Examples
- 22 Arithmetic Sequences: Definition & Finding the Common Difference – Video & Lesson Transcript
- 23 Finding the Common Difference
- 24 The Formula
- 25 Arithmetic Sequences and Series
- 26 Arithmetic Series
- 27 Arithmetic Sequence Calculator
- 28 What is an arithmetic sequence?
- 29 Arithmetic sequence definition and naming
- 30 Arithmetic sequence examples
- 31 Arithmetic sequence formula
- 32 Difference between sequence and series
- 33 Arithmetic series to infinity
- 34 Arithmetic and geometric sequences
- 35 Arithmetico–geometric sequence
- 36 Arithmetic sequence calculator: an example of use
- 37 Arithmetic Sequences and Sums
- 38 Arithmetic Sequence
- 39 Advanced Topic: Summing an Arithmetic Series
- 40 Footnote: Why Does the Formula Work?

## How do you find the common difference in arithmetic sequences?

Therefore, you can say that the formula to find the common difference of an arithmetic sequence is: d = a(n) – a(n – 1), where a(n) is the last term in the sequence, and a(n – 1) is the previous term in the sequence.

## What is the common difference arithmetic?

The constant difference between consecutive terms of an arithmetic sequence is called the common difference. Example: To find the common difference, subtract any term from the term that follows it. d=7−9=−2. −2 is the common difference between the terms.

## What is the example of common difference?

If the difference between every pair of consecutive terms in a sequence is the same, this is called the common difference. For example, the sequence 4,7,10,13, has a common difference of 3. A sequence with a common difference is an arithmetic progression.

## Whats a common difference?

Definition of common difference: the difference between two consecutive terms of an arithmetic progression.

## What is the common difference in the sequence 3?

The common difference is the difference between the numbers in a sequence. In the sequence 3,6,9,12, and 15 the common difference is 3.

## What is the common difference of the arithmetic sequence 3 13 4?

What is the common difference of the arithmetic sequence 3 13 4? The common difference in the arithmetic sequence 3, 13/4, 7/2, 15/4 is ¼. To find the difference, use the formula: d = a₂ – a₁.

## Common Difference – Formula, How to Find Common Difference?

A notion that is employed in sequences and arithmetic progressions is the concept of common difference. The commemoration of people’s birthdays might be regarded one of the many examples of sequence that can be observed in everyday life. When comparing successive festivities of the same individual, one year is the most typical variance between the celebrations. In this post, we will discuss the concept of common difference and how to identify it via the use of solved cases.

## What Is Common Difference?

An arithmetic sequence progresses from one term to the next by always adding (or removing) the same amount from the previous term’s value. In arithmetic, the number that is added (or subtracted) at each stage of a series is referred to as the “common difference,” since if we subtract (that is, if we discover the difference of) subsequent terms, we will always receive this same value in the process. The letter “d” is most frequently used to signify the common difference. Consider the following arithmetic sequence: 2,4,6,8, and so on.

## Common Difference Formula

The value between each subsequent number in an arithmeticsequence is referred to as the common difference. So the formula for determining the common difference of an arithmetic series is: d = (a(n) – (a(n-1)), where (a(n) is the final term in a sequence and (a(n-1) is the prior term in a sequence. There are two types of arithmetic sequences: arithmetic sequences that are repeated, and arithmetic sequences that are not repeated.

- It is possible to increase or decrease the arithmetic sequence.

Some sequences are simply composed of random numbers, but others are composed of a predetermined pattern that is followed in order to arrive at the sequence’s terms. For example, the arithmetic sequence (or progression) is based on the addition of a constant number in order to reach the next term in the series. The number of terms that are added to each other remains constant (always the same). a 1, (a 1+ d), (a 1+ 2d), (a 1+ 3d), d denotes the common difference, which refers to the fact that the difference between two consecutive words provides the constant value that has been added to the total amount.

When arithmetic sequences are plotted on graphs, they have a linear character (as a scatter plot).

When looking at the graph above, the x-axis increases by a constant value of one, but the y-axis increases by a constant value of three.

## Finding Common Difference in Arithmetic Progression

Allow me to demonstrate by creating an arithmetic series with a beginning number of 2 and a common difference of 5 as an example.

- Our first term will be represented by the number one: 2. Our second term will be equal to the first term (2) plus the common difference (5), resulting in a second term of 7, as follows: As a result, our first two words in our series are 2 and 7. Our third term will be equal to the second term (7) plus the common difference (5), resulting in a third term of 12
- So, the first three terms of our series are 2, 7, and 12
- And the first three terms of our sequence are 2, 7, and 12. Our fourth term will be equal to the third term (12) plus the common difference (5), resulting in a fourth term of 17
- Thus, the first four terms of our sequence are 2, 7, 12, and 17
- The first four terms of our sequence are 2, 7, 12, and 17
- And the first four terms of our sequence are 2, 7, 12, and 17

The process of creating an arithmetic series from a beginning number and one common difference is now familiar to us.

Let’s have a look at how to determine the common difference between two sequences. 2,7,12,.

- In this case, the first term is 2
- Hence, 2 is the starting number
- The second term is 7. For the difference between this and the first term, we divide 7 by 2 to get 5 as the answer. Consequently, there is a 5 point differential between the first and second terms
- The second term is 7 points, and the third term is 12 points. To get the difference, we divide 12 by 7, which results in a result of 5. As a result, the average difference between each word is 5

The following table contains some other instances of arithmetic sequences, as well as instructions on how to calculate the common difference of the series.

Arithmetic Sequence | Common Difference ‘d’ |

1, 6, 11, 16, 21, 26,. | d = 5. 5 is added to each term to arrive at the next term. so. the difference a2 – a1 = 5. |

10, 8, 6, 4, 2, 0, -2, -4, -6,. | d = -2. -2 is added to each term to arrive at the next term.so. the difference a2 – a1 = -2. |

1, 1/2, 0, -1/2,. | d = -½. A -½ is added to each term to arrive at the next term.so. the difference a2 – a1 = -½. |

Articles Related to “Common Difference” Take a look at the pages below that are linked to Common Difference.

- Arithmetic Progression is a type of progression in mathematics. Calculator for the Arithmetic Sequence
- Calculator of Sequences

Important Points to Keep in Mind About the Common Difference Here is a list of some of the most crucial factors to consider while discussing common differences.

- In an arithmetic sequence, the same amount is always added or subtracted from each term as it moves from one to the next. It is referred to as the “common difference” when the same number is added or removed at each level of an arithmetic sequence. The letter ‘d’ stands for the common difference

## Examples on Common Difference

- Example 1: Identify the common difference between the following sequences. 3,0,3,6,9,12,. Solution: Consider the following sequence: 3,0,3,6,9,12,. The distance between each number in the series is the common point of differentiation. Take note of the fact that each number is three numbers apart from the preceding number. Twelve nines (39 sixes) equals thirty-three threes (33 zeros) equals thirty-three threes (three threes). As a result, the usual difference is three. Example 2: What is the one thing that all of the following sequences have in common? 3,11,19,27,35. Solution: The following numbers are in the given sequence: 3, 11, 19, 27, and 35. To determine the common difference, just subtract the first term from the second term, or the second term from the third term, or the second term from the fourth term, and so on. 81911 = 113 = 81911 = 8 Clearly, we are increasing the number by 8 each time we reach the following phrase. As a result, the average difference is 8.

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

## FAQs on Common Difference

When the difference between two consecutive terms remains constant, this is referred to as an arithmetic sequence. The common difference is the difference between two successive words that is used in a sentence.

### How To Find the Common Difference?

The common difference between each number in an arithmetic series is the value of the difference between them.

- Step 1: Choose any two terms that are consecutive
- In Step 2, find out what their difference is, which is denoted by D = (n-1), where N is the length of the sequence, and a(n-1) is the length of the preceding term in the sequence

### Can the Common Difference of AP Be Negative?

Yes, the common difference of an arithmetic progression might be positive, negative, or even zero depending on the situation at hand. A declining arithmetic series always has a positive common difference since the sequence starts off negative and continues to descend.

### Can the Common Difference in AP Be Zero?

Yes. In an arithmetic progression, the most common difference might be equal to zero. If the difference between consecutive terms is constant, then a sequence of terms is regarded to be an arithmetic progression, according to the definition of an arithmetic progression (AP). As a result, an AP might have a common difference of zero. Furthermore, because 0 is a constant, it should be included as a common difference; nonetheless, it appears to be a little out of place for all of the numbers to be equal while being in an arithmetic sequence.

### What Is the Symbol of Common Difference?

The common difference is the difference in value between each term in an arithmetic sequence, and it is symbolized by the symbol ‘d.’ It is the value between each term in an arithmetic sequence.

### What If the Common Difference Is Not Constant?

If the common difference between subsequent words in a sequence does not remain constant throughout time, the sequence cannot be termed mathematical in nature.

### What Is the Common Difference Formula?

For an arithmetic series, the common difference may be calculated using the formula: d = a(n), where the final term in the sequence is a(n – 1), and the prior term in the sequence is a(n – 1).

## Finding Common Differences

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

### A General Note: Arithmetic Sequence

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

The common difference is the name given to this constant. If is the initial term of an arithmetic series anddis the common difference, the sequence will be as follows:left right=left , +d, +2d, +3d,.left right=left , +d, +2d, +3d,.right

### Example 1: Finding Common Differences

Is each of the sequences mathematical in nature? If this is the case, identify the common difference.

### Solution

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

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

### Analysis of the Solution

Figure 1 depicts the graph of each of these sequences. Figure 1: Graph of each series Observe that, while both sequences indicate development, an is not linear whereas bis is linear, as shown by the graphs. Because arithmetic sequences evolve at a constant pace, their graphs will always consist of points on a line (as seen in Figure 1).

### QA

No. As long as we know that the sequence is arithmetic, we may take any one term from it and subtract it from the following term to determine the common difference.

### Try It 1

Is the provided sequence a logical sequence? If this is the case, identify the common difference. 16 dots on the left, 14 dots on the right, 12 dots on the left, 10 dots on the right, 16 dots on the right

### Try It 2

Is the provided sequence a logical sequence? If this is the case, identify the common difference. 3, text 6, text 10, text 15, dots on the left, solution on the right

## Writing Terms of Arithmetic Sequences

After recognizing an arithmetic sequence, we can determine the terms if we are provided the first term as well as the common difference between the two terms. The terms may be discovered by starting with the first term and repeatedly adding the common difference to the end of the list. Furthermore, any term may be obtained by putting the values ofnanddin into the formula below, which can be found in the table below. +left(n – 1)d +left(n – 1)d +left(n – 1)d

### How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.

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

### Example 2: Writing Terms of Arithmetic Sequences

Fill in the blanks with the first five terms of the arithmetic sequence beginning with_ =17 andd=-3.

### Solution

The operation of adding three is the same as the operation of subtracting three. Starting with the first phrase, remove 3 from each word to arrive at the next term in the sequence. The first five terms are still available.

### Analysis of the Solution

In keeping with expectations, the graph of the series is composed of points on a line, as seen in Figure 2. Fig. 2: A diagrammatic representation of a diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation of the diagrammatic representation

### Try It 3

Solution: Write out the first five terms of the arithmetic series beginning with_ =1 andd=5.

### How To: Given any the first term and any other term in an arithmetic sequence, find a given term.

- In order to solve ford, substitute the values provided for , ,ninto the formula = +left(n – 1right)d
- This will give you the answer ford. Calculate the value of a given term by substituting the necessary values for , n, anddinto the formula = +left(n – 1right)d.

### Example 3: Writing Terms of Arithmetic Sequences

Find_ if_ =8 and_ =14 are provided.

### Solution

The sequence may be stated in terms of the first term, 8, and the common differenced. Our fourth term equals 14; we know the fourth term has the pattern +3d=8+3d; we know the fourth term equals 14. We can determine the common differences between the two. begin = +left begin (n – 1 to the right) dhfillhfill = +3dhfillhfill =8+3dhfilltext14 d.hfill =8+3dhfilltext14 d.hfill =8+3dhfilltext14 d.hfill =8+3dhfilltext14 d.hfill =8+3dhfilltext14 d.hfill =8+3dhfill Find the fifth term by multiplying the common difference by the fourth term in the previous equation.

### Analysis of the Solution

Observe how each term’s common difference is multiplied by one in order to identify the following terms: once to find the second term, twice to get the third term, and so on. The tenth term might be determined by adding the common difference to the first term nine times, or by using the equation = +left(n – 1right)d to get the common difference between the first and ninth terms.

### Try It 4

Find .Solution for the variables_ =7 and_ =17.

## Introduction to Arithmetic Progressions

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

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

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

### Terminology and Representation

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

### General Form of an AP

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

### Sum of n Terms of Arithmetic Progression

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

### Derivation of the Formula

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

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

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

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

### Sample Problems on Arithmetic Progressions

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

Problem 2.

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

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

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

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

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

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

## Examples

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

Alternatively, there might be alternative sequences that begin in the same way and end in 100, but which do not contain the natural numbers less than or equal to 100. As a result, it would be advisable to express the sequence in such a way that the reader can grasp the overall pattern.

## Finding Patterns

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

#### Example 1

Examine a few finite sequences to see if any patterns can be discovered.

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

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

- There are 9fives in all, and the aggregate is 9 x 5 = 9.
- expand more Due to the fact that the difference between each term is constant, this sequence from 1 through 1000 is arithmetic.
- The first phrase, a1, is one and the last term, is one thousand thousand.
- The sum of all positive integers up to and including 1000 is 500 500.

## Common Difference: Formula & Overview – Video & Lesson Transcript

The amount between each integer in an arithmetic sequence is the most often encountered difference. It is referred to as a common difference because it is the same, or common to, all of the numbers in the series, as well as the difference between each of those numbers. To get the common difference between two numbers, you may simply subtract each number from the number that comes after it in the sequence. For example, what is the common difference between the numbers in the following sequence: 0 through 9?

Maintain the pattern by subtracting from each number in the series to guarantee that each number has the same pattern: 7 minus 4 equals 3 4 minus 1 equals 3.

You may state that the formula to get the common difference of an arithmetic series is_d=a (n) – a (n-1), wherea (n) is the last phrase in the sequence and a (n- 1) is the prior term in the sequence, and a (n- 1) is the previous term in the sequence.

## Examples

1.

What is the one thing that all of the following sentences have in common? 18 minus 13 equals 5 13 minus 8 equals 5 8 minus 3 equals 5

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

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

and so on are the numbers.

## Finding the Common Difference

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

- Take a look at it!
- Repeating this process with the 3 and the 4 will reveal that it too has a difference of 1, indicating that this arithmetic sequence has one common difference of one.
- When we deduct the 5 from the 8 and the 8 from the 11, we get a total of 3 as well.
- In this case, we obtain 2 if we remove 3 from 5 and 5 from 7 respectively.
- As a result, the most common difference for this sequence is number 2.

## The Formula

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

The common difference must be added once to the first term to get to the second term.## Arithmetic Sequences 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 all possible arithmetic means between the given terms: a1=3anda6=17
- A1=5anda5=7
- A2=4anda8=7
- A5=12anda9=72
- A5=15anda7=21
- A6=4anda11=1
- A7=4anda11=1

### Part B: Arithmetic Series

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

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

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

### Part C: Discussion Board

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

### Answers

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

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

## Arithmetic 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. When the same thing appears more than once in a sequence, this is known as a repetition. An arithmetic sequence is likewise a collection of objects — in this case, a collection of numbers.

Such a sequence can be finite if it has a predetermined number of terms (for example, 20), or infinite if we do not specify the number of terms.Each arithmetic sequence is uniquely defined by two coefficients: the common difference and the first term.Each arithmetic sequence is uniquely defined by two coefficients: the common difference and the first term.

## Arithmetic sequence definition and naming

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

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

## Arithmetic sequence examples

The following are some instances of an arithmetic sequence:

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

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

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

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

## Arithmetic sequence formula

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

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

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

## Difference between sequence and series

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

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

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

## Arithmetic series to infinity

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

## Arithmetic and geometric sequences

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

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

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

## Arithmetico–geometric sequence

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

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

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

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

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

## Arithmetic sequence calculator: an example of use

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

- However, we are only concerned with the distance traveled from the fifth to the ninth second of the second.
- Simply remove the distance traveled in the first four seconds (S4,) from the partial total S9.
- S4 = n/2 *= 4/2 *= 74.8 m = n/2 *= 4/2 *= S4 is the same as 74.8 meters.
- It is possible to use the arithmetic sequence formula to compute the distance traveled in each of the five following seconds: the fifth, sixth, seventh, eighth, and ninth seconds.
- Make an attempt to do it yourself; you will quickly learn that the outcome is precisely the same!

## 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:

- 1, 4, 7, 10, 13, 16, 19, 22, and 25 are the first four digits of the number 1. Has:

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:

- For example, consider the following values: a,d, andn

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: