Sum of Arithmetic Sequence Formula

We can calculate the sum of all terms in an arithmetic sequence using the sum of the arithmetic sequence formula.

The formula for the sum of the n terms of an arithmetic series when the last term is not given is

The formula for Sum When Last Term is Given:

The formula for the sum of the first n terms of an arithmetic sequence is

S_n = \frac{n}{2} \cdot \left( 2a + (n - 1)d \right)

If we write 2a as a + a, the formula becomes

S_n = \frac{n}{2} \cdot \left( a + a + (n - 1)d \right)

Recognizing that a + (n − 1)d = a_n​, we get:

S_n = \frac{n}{2} \cdot (a + a_n)

Where:

• S_n​ is the sum of the first n terms.
• a is the first term.
• a_n is the last term.
• n is the number of terms.

This formula is useful when the last term (a_n) is given.

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_n = 240, d = 2 and n = 12.

Use the formula S_n = n/2 [2a + (n - 1)d] to find the required value.
=> 240 = 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 the 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

Related Reads:

• Arithmetic Sequence
• Geometric Progression
• Harmonic Progression
• Arithmetic Sequence vs Geometric Sequence

Practice Problem Based on Sum of Arithmetic Sequence Formula

Question 1. An arithmetic sequence has a sum of 350 after 20 terms, and the common difference is 5. Find the first term of the sequence.

Question 2. Find the sum of an arithmetic sequence with the first term a=3, the last term aₙ=39, and the number of terms n=19.

Question 3. The sum of the first 15 terms of an arithmetic sequence is 570. The first term is 10. Find the common difference.

Question 4. The first term of an arithmetic sequence is a=12, the common difference is d=3, and the sum of the sequence is 300. How many terms are in the sequence?

Answer:-

1. -30.
2. 399.
3. 4.
4. 12.