Arithmetic Sequence Formula
The arithmetic sequence formula is used for the calculation of the nth term of an arithmetic progression. The arithmetic sequence is the sequence where the common difference remains constant between any two successive terms. If we want to find any term in the arithmetic sequence then we can use the arithmetic sequence formula. Let us understand the arithmetic sequence formula using solved examples.
What is the Arithmetic Sequence Formula?
For the calculation using the arithmetic sequence formulas, it is important to know the 1st term of the sequence, the number of terms, and the common difference. There are different formulas associated with an arithmetic series used to calculate the nth term, sum, or the common difference of a given arithmetic sequence.
Formula 1: The arithmetic sequence formula is given as,
\(a_n = a_1 + (n  1)d\)
where,
 \(a_n\) = nth term,
 \(a_1\)_{ }= first term, and
 d is a common difference
The above formula is also referred to as the nth term formula of an arithmetic progression.
Formula 2: The sum of first n terms in an arithmetic sequence is given as,
\( S_n=\frac{n}{2}[2 a+(n1) d] \)
where,
 \(S_n\) = Sum of n terms,
 \(a_1\)_{ }= first term, and
 d is a common difference
Formula 3: The formula for calculating the common difference of an AP is given as,
\(d = a_n−a_{n1}\)
where,
 \(a_n\) = nth term,
 \(a_{n1}\)_{ }= second last term, and
 d is a common difference
Formula 4: The sum of first n terms of an arithmetic progression when the nth term, \(a_n\) is known is given as,
\( S_n=\frac{n}{2}[a_1+a_n] \)
where,
 \(S_n\) = Sum of first n terms
 \(a_n\) = nth term, and
 \(a_1\)_{ }= first term
Let us have a look at a few solved examples to understand the arithmetic sequence formula better.

Example 1: Using the arithmetic sequence formula, find the 13th term in the sequence 1, 5, 9, 13...
Solution:
To find: 13th term of the given sequence.
Since the difference between consecutive terms is same, the given sequence forms arithmetic sequence.
a = 1, d = 4
Using arithmetic sequence formula
\(a_n = a_1 + (n  1)d\)
For 13^{th} term, n = 13
\(a_n\) = 1 + (13  1)4
\(a_n\) = 1 + (12)4
\(a_n\) = 1 + 48
\(a_n\) = 49
Answer: 13th term in the sequence is 49.

Example 2: Find the first term in the arithmetic sequence where the 35th term is 687 and the common difference 14.
Solution:
To find: The first term in the arithmetic sequence
Given:
\(a_n\) = n^{th} term, d = 14
Using Arithmetic Sequence Formula
\(a_n = a_1 + (n  1)d\)
687 = \(a_1\)_{ }+ (35  1)14
687 = \(a_1\)_{ }+ (34)14
687 = \(a_1\)_{ } + 476
\(a_1\) = 211
Answer: The first term in the sequence is 211.