Sum of Arithmetic Sequence Formula
Before we begin to learn about sum of 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.
Formula to Find Sum of Arithmetic Sequence
Consider an arithmetic sequence (AP) whose first term is \(a_1\) (or) \(a\) and the common difference is \(d\).
Formula 1: The sum of first n terms of arithmetic sequence where \(n^\text{th}\) term is not known is given by:
\[S_{n}=\frac{n}{2}[2a+(n1)d]\]
Formula 2: The sum of first n terms of the arithmetic sequence where \(n^\text{th}\) term, \(a_n\) is known is given by:
\[S_{n}=\frac{n}{2}[a_1+a_n]\]
Solved 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 \(n=10\).
\(\begin{align}S_{n}&=\frac{n}{2}[2a_1+(n1)d]\\&=\frac{10}{2}[2(4)+(101)3]\\&=5 \times (8 + 27)\\&=95\end{align}\)
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 \(a_7=40\).
\(\begin{align}S_{n}&=\frac{n}{2}[a_1+a_n]\\&=\frac{7}{2}[10+40]\\&=\frac{7\times 50}{2}\\&= 7 \times 25 \\ &= 175 \end{align}\)
Answer: Sum of 7 terms of the given arithmetic sequence = 175.