Arithmetic Sequence Explicit Formula
Before going to learn the arithmetic sequence explicit formula, let us recall what is an arithmetic sequence. It is a sequence of numbers in which the differences between every two consecutive numbers are all same. For example, 3, 6, 9, 12, ... is an arithmetic sequence as the difference between every term and its previous term is 3. This difference is usually known as the common difference and is denoted by d. The first term of an arithmetic sequence is represented by \(a_1\) or a. Let us learn the arithmetic sequence explicit formula along with a few solved examples.
What is Arithmetic Sequence Explicit Formula?
The arithmetic sequence explicit formula is used to find any term of the arithmetic sequence, \(a_1, a_2, a_3, ..., a_n,....\) using its first term and the common difference. The arithmetic sequence explicit formula is:
\(a_n=a+(n1)d\)
where,
 \(a_n\) = n^{th} term of the arithmetic sequence
 a = the first term of the arithmetic sequence
 d = the common difference (the difference between every term and its previous term. i.e., \(d=a_na_{n1}\)
Let us work on a few solved examples to learn the arithmetic sequence explicit formula better.
Solved Examples Using Arithmetic Sequence Explicit Formula

Example 1: Find the 15^{th} term of the arithmetic sequence 3, 1, 1, 3, ..... using the arithmetic sequence explicit formula.
Solution:
The first term of the given sequence is, a = 3.
The common difference is, d = 1  (3) (or) 1  (1) (or) 3  1 = 2.
The 15^{th} term of the given sequence is calculated using,
\(a_n=a+(n1)d\)
\(a_{15}\) = 3 + (15  1) 2 = 3 + 14(2) = 3 + 28 = 25.
Answer: The 15^{th} term of the given sequence = 25.

Example 2: Find the general term (or) n^{th} term of the arithmetic sequence 1/2, 2, 9/2, .....
Solution:
The first term of the given sequence is, a = 1/2.
The common difference is d = 2  (1/2) (or) 9/2  2 = 5/2.
We can find the general term (or) n^{th} term of an arithmetic sequence using the arithmetic sequence explicit formula.
\(a_n=a+(n1)d\)
\(a_n=  \dfrac{1}{2}+(n1) \dfrac{5}{2}\)
\(a_n =  \dfrac{1}{2} + \dfrac{5}{2} n  \dfrac{5}{2}\)
\(a_n = \dfrac{5}{2} n  3\)
Answer: The n^{th} term of the given sequence = (5/2) n  3.