Sum of Arithmetic Sequence Formula

What is an Arithmetic Sequence?

In mathematics, many people are already familiar with common operations such as addition, subtraction, multiplication, and division. Throughout primary and secondary school, students become well-versed with adding numbers together or subtracting them from one another.

Sometimes, students are even asked to add a whole sequence of numbers together. Some terms to learn for this lesson include:

• Sequence: a series of numbers placed in a specific order.
• Arithmetic sequence: a special type of sequence. An arithmetic sequence, also known as an arithmetic progression or an arithmetic series, is a set of numbers in which the difference between each term is a constant value. In other words, the same value can be added to the previous number for an infinite amount of time.
• Arithmetic series: another name for an arithmetic sequence.

It’s a relatively straightforward process to add all of the terms together in a short series to derive the sum of an arithmetic sequence. However, what if a student is asked to add the first 20 or first 125 terms of an arithmetic sequence? What if the difference between each term is an especially large value? Is there an easier way to solve this type of problem?

Using the general formula for the nth term of an arithmetic sequence, an individual can easily find the sum of an arithmetic series for a specific number of terms.

Arithmetic Sequence Examples

An arithmetic sequence can be expressed in terms of the following expression: a,(a+d),(a+2d),(a+3d), … where a is

the first term and d is the constant difference between values.

Using this expression, some arithmetic sequence examples include:

• 1, 5, 9, 13, 17, 21, 25, 29, 33, …
  • The constant value can be derived by taking the difference between any two adjacent terms. For example, 9 – 5 = 4 and 33 – 29 = 4. Therefore, 4 represents the constant value.
  • Using the expression above: 1, (1 + 4) , (1 + 2(4)), (1 + 3(4)), (1 + 4(4)),…
• 2, 4, 6, 8, 10, 12, 14, 16, 18,…
  • In this simple arithmetic series, the constant value between adjacent terms is 2.
  • Using the expression above: 2, (2 + 2), (2 + 2(2)), (2 + 3(2)), (2 + 4(2)),…
• 1, 8, 15, 22, 29, 36, 43, 50, …
  • The constant value can be derived by taking the difference between any two adjacent terms. In this sequence, 50 – 43 = 7 and 22 – 15 = 7. Therefore, the constant value is 7.
  • Using the expression above: 1, (1 + 7), (1 + 2(7)), (1 + 3(7)), (1 + 4(7)),…
• 5, 15, 25, 35, 45, 55, 65, 75, …
  • Taking the difference between any two adjacent terms, the constant value is 10. For example, 15 – 5 = 10 and 55 – 45 = 10.
  • Using the expression above: 5, (5 + 10), (5 + 2(10)), (5 + 3(10)), (5 + 4(10)),…
• 12, 24, 36, 48, 60, 72, 84, 96, …
  • The constant value for this sequence is 12. This is derived by subtracting any two adjacent terms, such as 24 – 12 = 12 or 84 – 72 = 12.
  • Using the expression above: 12, (12 + 12), (12 + 2(12)), (12 + 3(12)), (12 + 4(12)),…

Sum of an Arithmetic Sequence

Calculating the sum of the first few terms of an arithmetic series is relatively simple. These values just simply need to be added together. However, the addition of the first 100 terms, for example, would prove more problematic. Rather than manually adding all of these terms together, mathematicians have developed the arithmetic sequence formula, which allows an individual to easily calculate the sum of an arithmetic sequence.

The formula for the sum of an arithmetic sequence is:

• n = the number of terms to be added
• a = the first term in the sequence
• d = the constant value between terms

For example, adding the first four terms of the arithmetic sequence 2, 4, 6, 8,… can be manually calculated by adding the terms together: 2 + 4 + 6 + 8 = 20. Using the formula:

Arithmetic Sequence

Some numbers have patterns to them such as 2, 4, 6, 8, 10. The pattern here is that each successive number is two (2) more than the previous number. Say we wanted to sum all of the numbers in this pattern from 2 to 200. That would take a long time to list out the values. Luckily there is an equation to simplify this. Let’s practice using this equation.

Problems for Additional Practice

1. Add up the first four terms of the arithmetic sequence {2, 4, 6, 8, 10}.

2. Add up the first ten terms of the arithmetic sequence {100, 200, 300, …}.

3. Add up the first 1,000 terms of the arithmetic sequence {3, 6, 9, …}.

4. A bank wants to do a promotion. They have provided a puzzle and if it is solved correctly the winner gets the money in a savings account in their bank. The puzzle is is this. A triangle has 20 rows. The first row is $1. Each successive row has one more dollar. How much money is in the triangle?

Solutions

For each solution we will use the arithmetic sum equation (n/2)(2a + (n – 1)d), where a is the first term, d is the common difference between the terms and n is how many terms to sum.

1. (4/2)(2(2) + (4 – 1)(2)) = 2(4 + (3)(2)) = 2(4 + 6) = 2(10) = 20

To check ourselves we can add 2 + 4 + 6 + 8 and we get 20.

2. (10/2)(2(100) + (10 – 1)(100)) = 5(200 + (9)(100)) = 5(200 + 900) = 5(1100) = 5,500

3. (1,000/2)(2(3) + (1,000 – 1)(3)) = 500(6 + (999)(3)) = 500(5 + 2,997) = 500(3,002) = 1,501,000

4. (50/2)(2(1) + (50 – 1)(1)) = 25(2 + 49) = (25)(51) = $1,275

Sum of Arithmetic Sequence Formula

Before we begin to learn about sum of the arithmetic sequence formula, let us recall what is an arithmetic sequence. An arithmetic sequence is a sequence of numbers where each successive term is a sum of its preceding term and a fixed number. This fixed number is called a common difference. So, in an arithmetic sequence, the differences between every two consecutive terms are the same.

Let us learn the sum of arithmetic sequence formula with a few solved examples.

What Is the Sum of Arithmetic Sequence Formula?

The sum of the arithmetic sequence formula is defined as the formula to calculate the total of all the terms present in an arithmetic sequence. We know that an arithmetic series of finite arithmetic progress follows the addition of the members which is given by (a, a + d, a + 2d, …) where “a” = the first term and “d” = the common difference.

Sum of Arithmetic Sequence Formula

Consider an arithmetic sequence (AP) whose first term is a and the common difference is d.

Formula 1: The sum of first n terms of an arithmetic sequence where nth term is not known is given by:

Where

• Sn

 = the sum of the arithmetic sequence, a  = the first term, d = the common difference between the terms, n = the total number of terms in the sequence and an

•   = the last term of the sequence.

Formula 2: The sum of first n terms of the arithmetic sequence where nth term, Sn is known is given by:

Where

• $S_n$= the sum of the arithmetic sequence,
• $a_1$= the first term,
• $d$= the common difference between the terms,
• $n$= the total number of terms in the sequence and
• $a_n$= the last term of the sequence.

Derivation of Sum of Arithmetic Series Formula

In an arithmetic sequence is a sequence, every term after the first is obtained by adding a constant, referred to as the common difference (d).

Examples Using Sum of Arithmetic Sequence Formula

Example 1: Find the sum of arithmetic sequence -4, -1, 2, 5, … up to 10 terms.

Solution:

Here, $a_1=-4$

and $n=10$

.Using the sum of arithmetic sequence formula,

Answer: Sum of arithmetic sequence -4, -1, 2, 5, … up to 10 terms = 95.

Example 2: Find the sum of 7 terms of an arithmetic sequence whose first and last terms are 10 and 40 respectively.

Solution:

Here, $a_1=10$

and $a_7=40$

.Using the sum of arithmetic sequence formula,

Answer: Sum of 7 terms of the given arithmetic sequence = 175.

Example 3: Using the sum of arithmetic sequence formula, calculate the sum of the first 20 terms of the sequence 1, 5, 9, 13, ……

Solution:

Here, $a_1=1$

, $d=4$ and $n=20$

Using the sum of arithmetic sequence formula,

Answer: Sum of arithmetic sequence 1, 5, 9, 13, …… = 780.

 term is not known:

The sum of first n terms of the arithmetic sequence where $n^{\text{th}}$ term, $a_n$ is known:

How To Use the Sum of Arithmetic Sequence Formula?

•  = the last term.
• Step 2: Put the given values in the appropriate formula,

How To Derive the Sum of Arithmetic Sequence Formula?

 = a${}_1$ + (n – 1)d.

• Step 1: Find the first term
• Step 2: Check for the number of terms.
• Step 3: Generalize the formula for the first term, that is a₁ and thus successive terms will be a₁+d, a₁+2d
• Step 4: Find the last term, $a_n=a_1+(n-1)d$
• Step 5: Find their sum. (first, by adding the terms successively and second, by combining the terms and successively subtracting the common difference)
• Step 6: Subtracting the two equations, we get the formula as 

What Is a_n In the Sum of Arithmetic Sequence Formula?

, $a_n$ refers to the $n^{\text{th}}$ term of the given arithmetic sequence.

Formula for Sum of Arithmetic Sequence Formula

There are two ways with which we can find the sum of the arithmetic sequence. The formulas for the sum of the arithmetic sequence are given below:

Notations:

• “S” is the sum of the arithmetic sequence,
• “a” as the first term,
• “d” the common difference between the terms,
• “n” is the total number of terms in the sequence and
• “L” is the last term of the sequence.

Solved Example Using Sum of Arithmetic Sequence Formula

Question: Find the sum of the first 30 terms of the sequence 1, 3, 5, 7, 9 ……

Solution:

Given,

a = 1
d = 2
n = 30

Using the formula: S = n/2 {2a + (n − 1)d}

S = 30/2 {2(1) + (30 − 1)2}

= 900

where:

• a is the first term, and
• d is the difference between the terms (called the “common difference”)

Rule

We can write an Arithmetic Sequence as a rule:

x_n = a + d(n−1)

(We use “n−1” because d is not used in the 1st term).

So the 9th term is:

x₉ = 5×9 − 2
= 43

Advanced Topic: Summing an Arithmetic Series

To sum up the terms of this arithmetic sequence:

a + (a+d) + (a+2d) + (a+3d) + …

use this formula:

 

Here is how to use it:

 

Footnote: Why Does the Formula Work?

Let’s see why the formula works, because we get to use an interesting “trick” which is worth knowing.

First, we will call the whole sum “S”:

Sum of Arithmetic Sequence Formula

A sequence is an arrangement of any things or a group of numbers in a certain order that follows a rule. Basically, it is a set of numbers (or items) that follow a specific pattern. For example, 5, 10, 15, 20…. is a sequence as every time the value is getting incremented by 5. If the sequence’s elements are in ascending order, the sequence’s order is ascending. If the sequence’s elements are in decreasing order, the sequence’s order is decreasing. Arithmetic sequence, geometric sequence, Fibonacci sequence, harmonic sequence, triangular number sequence, square number sequence, and cube number sequence are a few examples of specific sequences.

Arithmetic Sequence

An arithmetic sequence is a number series in which each subsequent term is the sum of its preceding term and a constant integer. This constant number is referred to as the common difference. As a result, the differences between every two successive terms in an arithmetic series are the same.

If the first term of an arithmetic sequence is a and the common difference is d, then the terms of the arithmetic sequence are of the form:

 a, a+d, a+2d, a+ 3d, a+4d, ….

Suppose n is the total number of terms in the sequence.

For n = 1, the sequence is a.

For n = 2, the sequence is a, a + d.

For n = 3, the sequence is a, a + d, a + 2d.

For n = 4, the sequence is a, a + d, a + 2d, a + 3d.

Hence, the general term of the sequence is a_n = a + (n – 1)d.

Sum of the arithmetic sequence

The formula for calculating the sum of all the terms in an arithmetic sequence is defined as the sum of the arithmetic sequence formula. If an arithmetic sequence is written as in the form of addition of its terms such as, a + (a+d) + (a+2d) + (a+3d) + ….., then it is known as arithmetic series. The sum of the first n terms of an arithmetic series in which the nth term is unknown is given by:

S_n = n/2 [2a + (n – 1)d]

where,

S_n = sum of the arithmetic sequence,

a  = first term of the sequence,

d = difference between two consecutive terms,

n = number of terms in the sequence.

If we write 2a in the formula as (a + a), the formula becomes, S_n = n/2 [a + a + (n – 1)d]

We know, a + (n – 1)d is denoted by a_n. Hence, the formula becomes, S_n = n/2 [a + a_n]

Derivation

Suppose the first term of a sequence is a, common difference is d and the number of terms are n.

We know the n^th term of the sequence is given by, 

a_n = a + (n – 1)d         …… (1)

Also the sum of the arithmetic sequence is,

S_n = a + (a + d) + (a + 2d) + (a + 3d) + …… +  a + (n – 1)d     …… (2)

From (1), the equation (2) can also be expressed as,

S_n = a_n + a_n – d + a_n – 2d + a_n – 3d + …… +  a_n – (n – 1)d        …… (3)

Adding (2) and (3) we get,

2 S_n = [a + (a + d) + (a + 2d) + (a + 3d) + …… +  a + (n – 1)d] + [a_n + a_n – d + a_n – 2d + a_n – 3d + …… +  a_n – (n – 1)d]

2 S_n = (a + a + a + ….. n times) + (a_n + a_n + a_n + ….. n times)

2 S_n = n (a + a_n)

S_n = n/2 [a + a_n]

This derives the formula for sum of an arithmetic sequence.

Sample Questions

Question 1. Find the sum of the arithmetic sequence: 4, 10, 16, 22, …… up to 10 terms.

Solution:

We have, a = 4, d = 10 – 4 = 6 and n = 10.

Use the formula S_n = n/2 [2a + (n – 1)d] to find the required sum.

S₁₀ = 10/2 [2(4) + (10 – 1)6]

= 5 (8 + 54)

= 5 (62)

= 310

Question 2. Find the sum of the arithmetic sequence: 7, 9, 11, 13, …… up to 15 terms.

Solution:

We have, a = 7, d = 9 – 7 = 2 and n = 15.

Use the formula S_n = n/2 [2a + (n – 1)d] to find the required sum.

S₁₅ = 15/2 [2(7) + (15 – 1)2]

= 15/2 (14 + 28)

= 15/2 (42)

= 315

Question 3. Find the first term of an arithmetic sequence if it has a sum of 240 for a common difference of 2 between 12 terms.

Solution:

We have, S = 200, d = 2 and n = 12.

Use the formula S_n = n/2 [2a + (n – 1)d] to find the required value.

=> 200 = 12/2 [2a + (12 – 1)2]

=> 240 = 6 (2a + 22)

=> 40 = 2a + 22

=> 2a = 18

=> a = 9 

Question 4. Find the common difference of an arithmetic sequence of 8 terms having a sum of 116 and the first term as 4.

Solution:

We have, S = 116, a = 4, n = 8.

Use the formula S_n = n/2 [2a + (n – 1)d] to find the required value.

=> 116 = 8/2 [2(4) + (8 – 1)d]

=> 116 = 4 (8 + 7d)

=> 29 = 8 + 7d

=> 7d = 21

=> d = 3

Question 5. Find the sum of an arithmetic sequence of 8 terms with first and last terms as 4 and 10 respectively.

Solution:

We have, a = 4, n = 8 and a_n = 10.

Use the formula S_n = n/2 [a + a_n] to find the required sum.

S₈ = 8/2 [4 + 10]

= 4 (14)

= 56

Question 6. Find the number of terms of an arithmetic sequence with the first term, last term and sum as 16, 12 and 140 respectively.

Solution:

We have, S = 140, a = 16 and a_n = 12.

Use the formula S_n = n/2 [a + a_n] to find the required value.

=> 140 = n/2 [16 + 12]

=> 140 = n/2 (28)

=> 14n = 140

=> n = 10

Question 7. Find the sum of an arithmetic sequence with the first term, common difference and last term as 8, 7 and 50 respectively.

Solution:

We have, a = 8, d = 7 and a_n = 50.

Use the formula a_n = a + (n – 1)d to find n.

=> 50 = 8 + (n – 1)7

=> 42 = 7 (n – 1)

=> n – 1 = 6

=> n = 7

Use the formula S_n = n/2 [a + a_n] to find the sum of sequence.

S₇ = 7/2 (8 + 50)

= 7/2 (58)

= 203

Sum of the First n Terms of an Arithmetic Sequence

Suppose a sequence of numbers is arithmetic (that is, it increases or decreases by a constant amount each term), and you want to find the sum of the first n terms.

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