Sum of Arithmetic Sequence can be calculated using the formula S_n = n/2 [2a+(n−1)d], where a is the first term of sequence and d is common difference.

Before learning how to find the sum of an arithmetic sequence, let’s learn about arithmetic sequences first.

What is Arithmetic Sequence?

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the “common difference.”

An arithmetic sequence is a sequence of numbers a₁, a₂, a₃, … such that for every n (where n is a positive integer): a_n+1 = a_n + d where:

• a_n​ is the n^th term of the sequence.
• d is the common difference between consecutive terms.

Formula for the Sum of an Arithmetic Sequence

S_n = n/2 [2a+(n−1)d],

OR

S_n = n/2 [a + a_n]

Where,

• n is number of terms for which we are finding sum,
• d is the common difference,
• a is first term of AP, and
• a_n is last term of AP.

Steps to Find Sum of an Arithmetic Sequence

To find the sum of an arithmetic sequence, you can use the formula for the sum of the first n terms of the sequence. Here are the steps to find the sum:

Step 1: Identify the First Term (a)

Step 2: Determine the Common Difference (d)

Step 3: Determine the Number of Terms (n) if not given.

Step 4: Find the n_th term(a_n) or Last Term (l).

Step 5: Use the Sum Formula.

Let’s consider an example for better understanding.

Example: Find the sum of the first 5 terms of the arithmetic sequence: 2, 5, 8, 11, 14.

Solution:

First term a = 2

Common difference d = 5 – 2 = 3

Number of terms n = 5

nth term an = 14

Thus, sum of given arithmetic sequence is S_n = n/2 [a + a_n]

⇒ S_n = 5/2 [2 + 14] = 5/2 × 16 = 5 × 8 = 40

Thus, the sum of the first 5 terms is 40.

Sample Examples on Sum of an Arithmetic Sequence

Example 1: Find the sum of the first 14 terms of the arithmetic sequence where the first term is 2 and the common difference is 3.

Solution:

Formula for finding sum of first n terms of AP is: S_n = n/2 [2a+(n−1)d]

Given: first term a = 2

Common difference d = 3

Number of terms n = 14

Thus, S_n = n/2 [2a + (n − 1) d] = 14/2 [ 2 × 2 +(14 – 1) 3] = 7 [4 + 13 × 3] = 7[43] = 301

The the sum of the first 14 terms of given arithmetic sequence is 301.

Example 2: If the sum of a given arithmetic sequence a = 2, d = 3 is 65. Find the number of terms.

Solution:

Given: a = 2 d = 3 Sn = 65

Thus, S_n = n/2 [2a+(n−1)d]

⇒ 65 = n/2 [ 2 × 2 + (n – 1) 3]

⇒ 65 = n/2 [ 4 + 3n – 3]

⇒ 130 = n(3n + 1)

⇒ 3n² + n – 130 = 0

Solving the above quadratic equation we will consider positive value 6 as answer.

Example 3: The sum of the first 20 terms of an arithmetic sequence is 500, and the common difference is 2. Find the first term.

Solution:

Given: n = 20, d = 3, S_n = 500

Thus, S_n = n/2 [2a+(n−1)d]

⇒ 500 = 20/2 [ 2a + (20 – 1) 2]

⇒ 500 = 10 [ 2a + 38]

⇒ 500 = 20a + 380

⇒ 120 = 20a

⇒ a = 6

Example 4: Find the sum of first 10 terms of arithmetic sequence with the first term 5, the last term 50.

Solution:

S_n = n/2 [a + an]

Where a is first term, an is last term and n is number of term.

Thus, S_n = 10/2 [ 5 + 50] = 5 × 55 = 275

Practice Problems on Sum of an Arithmetic Sequence

Problem 1: Find the sum of the first 20 terms of the arithmetic sequence where the first term a=3 and the common difference d=2.

Problem 2: Find the sum of the first 10 terms of an arithmetic sequence a=4 and d=−2.

Problem 3: The sum of the first 20 terms of an arithmetic sequence is 500, and the first term is 4. Find the common difference.

Problem 4: Find the sum of first 20 terms of arithmetic sequence with the first term 2, the last term 30.

Problem 5: The sum of the arithmetic sequence with a=5 and d=4 is 210. Find the number of terms.

Read More,

• Arithmetic Sequence
• Sum of Arithmetic Sequence Formula
• Geometric Sequence
• Harmonic Sequence