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 \(n^\text{th}\) term is not known is given by:

\[S_{n}=\frac{n}{2}[2a+(n-1)d]\]

Where

• \(S_{n}\) = 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
• \(a_{n}\)  = the last term of the sequence.

Formula 2: The sum of first n terms of the arithmetic sequence where \(n^\text{th}\) term, \(S_{n}\) is known is given by:

\[S_{n}=\frac{n}{2}[a_1+a_n]\]

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).

• Step 1: The \(n^\text{th}\) term of an arithmetic sequence, a\(_n\) = a\(_1\) + (n – 1)d, where the first term is a\(_1\), the second term is a\(_1\) + d, the third term is a\(_1\) + 2d, etc and this gives the formula of the sum of the arithmetic series, S\(_n\)

S\(_n\) = a\(_1\) + (a\(_1\) + d) + (a\(_1\) + 2d) + … + [a\(_1\) + (n–1)d] _____ eq(1)

• Step 2: We can also write it as, 

S\(_n\) = a\(_n\)​​​ + (a\(_n\) – d) + (a\(_n\)– 2d) + … + [a\(_n\) – (n–1)d] _____ eq(2) that is, the \(n^\text{th}\) term and successively subtracted the common difference

• Step 3: Add the above two equations together, we get

2S\(_n\) = n (a\(_1\) + a\(_n\)) ⇒ S\(_n\) = n(a\(_1\) + a\(_n\) )/2. Thus, S\(_n\) = n/2(a\(_1\) + a\(_n\)).

• Step 4: Substituting a\(_n\) = a\(_1\) + (n – 1)d, 

S_n = n/2 [a\(_1\) + a\(_1\) + (n – 1)d]  

Thus, S\(_n\) = n/2 [ 2a\(_1\) + (n – 1)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,

\(\begin{align}S_{n}&=\frac{n}{2}[2a_1+(n-1)d]\\&=\frac{10}{2}[2(-4)+(10-1)3]\\&=5 \times (-8 + 27)\\&=95\end{align}\)

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 \(a_7=40\).

Using the sum of arithmetic sequence formula,

\(\begin{align}S_{n}&=\frac{n}{2}[a_1+a_n]\\&=\frac{7}{2}[10+40]\\&=\frac{7\times 50}{2}\\&= 7 \times 25 \\ &= 175 \end{align}\)

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

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,

\(\begin{align}S_{n}&=\frac{n}{2}[2a_1+(n-1)d]\\&=\frac{20}{2}[2(1)+(20-1)4]\\&=10 \times (2 + 76)\\&=780\end{align}\)

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