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