**Sum of Arithmetic Sequence Formula**

Small Description: The formula for calculating the sum of all the terms that appear in an arithmetic sequence is referred to as the total of the arithmetic sequence formula.

This formula is defined as follows: We are aware that the addition of the series’ members, which is represented by the formula, is followed by an arithmetic series that has finite arithmetic progress.

Take into consideration an arithmetic sequence (AP) in which the first term is the letter a and the common difference is the letter d.

1. When the nth term of an arithmetic series is unknown, the following formula may be used to get the sum of the sequence’s first n terms:

Sn=(n/2)[2a+(n−1)d]

Where

Sn = 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

an = the last term of the sequence

2. If we know the nth term, Sn, then we may solve for the sum of the first n terms of the arithmetic series using the following formula:

Sn=(n/2)[a1+an]

Where

Sn = the sum of the arithmetic sequence,

- a1 = the first term,
- d = the common difference between the terms,
- n = the total number of terms in the sequence and
- an = the last term of the sequence.

**Solved Questions**

Example 1 : Find the sum of arithmetic sequence 8,3, -2, ….. Up to 20 terms.

Solution: Here a = 8 , d = 3 – 8 = -5, n = 20

Using the sum of an arithmetic sequence formula,

Sn = n / 2 [2a + (n – 1) d]

= 20 / 2 [2(8) + (20 – 1) -5]

= 10 [16 + 19(-5)]

= 10 [ 16 – 95]

= 10 x (-79)

= -790

Answer: Sum of arithmetic sequence 8,3,-2 …… = -790.

Example 2 : Find the sum of 9 terms of an arithmetic sequence whose first and last terms are 22 and 44 respectively..

Solution:

Here, a1=22 and a9=44

Using the sum of an arithmetic sequence formula,

Sn = n / 2 [a1 + an]

= 9 / 2 [22 + 44]

= 9 × (66/2)

= 9 × 33

= 297

Answer: Sum of 9 terms of the given arithmetic sequence = 297

Example 3: Using the sum of arithmetic sequence formula, calculate the sum of the first 24 terms of the sequence 5,7, 9,11, ……

Solution:

Here, a1 = 5 , d = 2 and n = 24

Using the sum of an arithmetic sequence formula,

Sn = n / 2 [2a1 + (n−1)d]

= 24 / 2 [2(5) + (24−1)2]

= 12× (10 + 46)

= 672

**Answer:** Sum of arithmetic sequence 5,7, 9,11, …… = 672