Recursive Formula
Before going to learn the recursive formula, let us recall what is a recursive function. A recursive function is a function that defines each term of a sequence using a previous term that is known, i.e. where the next term is dependent on one or more known previous term(s). A recursive function h(x) can be written as:
h(x) = a_{0}h(0) + a_{1}h(1) + ....... + a_{x1}h(x1) where a_{i }>= 0 and at least one of the a_{i} > 0
Let us learn the recursive formulas in the following section.
What Are Recursive Formulas?
Common examples of recursive functions are
1. n^{th} term of a Arithmetic Progression (A.P.)
a_{n }= a_{n1} + d for n ≥ 2
where
 a_{i} is the i^{th} term of a A.P.
 d is the common difference.
2. n^{th} term of a Geometric Progression (G.P.)
a_{n} = a_{n1}r for n ≥ 2
where
 a_{i} is the i^{th }term of a G.P.
 r is the common ratio.
a_{n} = a_{n1 }+ a_{n2} for n ≥ 2
Given that a_{0} = 1 and a_{1} = 1
where a_{i} is the i^{th }term of the sequence
Let us see the applications of the recursive formulas in the following section.
Solved Examples using Recursive Formula

Example 1: The recursive formula of a function is, f(x) = 5 f(x2) + 3, find the value of f(8). Given that f(0) = 0.
Solution.
f(8) = 5 f(6) + 3
f(6) = 5 f(4) + 3
f(4) = 5 f(2) + 3
f(2) = 5 f(0) + 3 = 3f(4) = 5 × 3 + 3 = 18
f(6) = 5 × 18 + 3 = 93
f(8) = 5 × 93 + 3 = 468Answer: The value of f(8) is 468

Example 2: Find the recursive formula for the following arithmetic sequence: 1, 6, 11, 16 .....
Solution.
Let a_{i} be the i^{th} term of the series and d be the common difference.
d = a_{2}  a_{1 }= 6  1 = 5
a_{n} = a_{n1} + 5
Answer: The recursive formula for this sequence is a_{n} = a_{n1} + 5