# 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) + ....... + \(a_{x-1}\) h(x-1) 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?

A recursive formula refers to a formula that defines each term of a sequence using the preceding term(s). The recursive formulas define the following parameters:

- The first term of the sequence
- The pattern rule to get any term from its previous term

### Recursive Formulas

Common examples of recursive functions are

Formula 1: \(n^\text{th}\) term of a Arithmetic Progression (A.P.)

\(a_{n}\) = \(a_{n-1}\) + d for n ≥ 2

where

- \(a_n\) is the \(n^\text{th}\) term of a A.P.
- d is the common difference.

Formula 2: \(n^\text{th}\) term of a Geometric Progression (G.P.)

\(a_n = a_{n-1} r \) for n ≥ 2

where

- \(a_n\) is the \(n^\text{th}\)
^{ }term of a G.P. - r is the common ratio.

Formula 3: Fibonacci Sequence

\(a_{n} = a_{n-1} + a_{n-2}\) for n ≥ 2

\(a_{0}\) = 1 and \(a_{1}\) = 1

where \(a_{n}\) is the \(n^\text{th}\)^{ }term of the sequence

Let us see the applications of the recursive formulas in the following section.

# Examples Using Recursive Formula

**Example 1: **The recursive formula of a function is, f(x) = 5 f(x-2) + 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 = 3

f(4) = 5 × 3 + 3 = 18

f(6) = 5 × 18 + 3 = 93

f(8) = 5 × 93 + 3 = 468

**Answer: **The value of f(8) is 468

**Example 2: **Find the recursive formula for the following arithmetic sequence: 1, 6, 11, 16 .....

**Solution.**

Let \(a_{n}\) be the \(n^\text{th}\) term of the series and d be the common difference.

\(d = a_{2} - a_{1}\)_{ }= 6 - 1 = 5

\(a_{n} = a_{n-1}\) + 5

**Answer:** The recursive formula for this sequence is \(a_{n} = a_{n-1}\) + 5

**Example 3:** The 13^{th} and 14^{th} terms of the Fibonacci sequence are 45 and 54 respectively. Find the 15^{th} term.

**Solution:**

Using Recursive formula for Fibonacci sequence,

15^{th} term is the sum of 13^{th} term and 14^{th} term.

15^{th} term = 13^{th} term + 14^{th} term

= 45 + 54

= 99

**Answer: **The 15^{th} term of the Fibonacci sequence is 99.

## FAQs on Recursive Formula

### What Is the Recursive Formula in Math?

A recursive formula is a formula that defines each term of a sequence using the preceding term(s)

### How To Find the Recursive Formula For a Sequence?

To find a recursive sequence in which terms are defined using one or more previous terms which are given.

- Step 1: Identify the \(n^\text{th}\) term of an arithmetic sequence and the common difference, d,
- Step 2: Put the values in the formula, \(a_{n}\) = \(a_{n-1}\) + d to find the (n+1)
^{th}term to find the successive terms.

### What Is the Recursive Formula For the Fibonacci series?

The Fibonacci series is characterized as the series in which each number is the sum of two numbers preceding it in the sequence. Thus, the Fibonacci formula is given as, \(F_n\) = F(n-1) + F(n-2), where n > 1.

### What Is \(a_n\) in Recursive Formula?

In any recursive formula, \(a_n\) refers to the \(n^\text{th}\) term in the sequence, which can be found using the recursive formulas:

- n
^{th}term of A.P: \(a_n\) = \(a_{n-1}\) + d for n ≥ 2 - n
^{th}term of G.P: \(a_n\)_{ }= \(a_{n-1}\) r for n ≥ 2 - Fibonacci Sequence: \(a_n\)
_{ }= \(a_{n-1}\)_{ }+ \(a_{n-2}\) for n ≥ 2

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