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Recursive Formula

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Recursion can be defined by two properties. A Base Case and Recursion Step. The base case is a terminating scenario that doesn’t use recursion to produce results. The recursion step consists of a set of rules that reduces the successive cases to forward the base case. A recursion or recursive formula is a formula that is used to tell us the next step in any recursion series. In a recursive series, each next term is dependent on the previous one or two terms.

In this article, we will learn about, Recursive Formulas or Recursion Formulas, Examples, and others in detail.

What Are Recursive Functions?

A recursive function is a function that defines each term of a sequence using the previous term i.e., The next term is dependent on one or more known previous terms. Recursive function h(x) is written as,

h(x) = a0h(0) + a1h(1) + a2h(2) + … + ax – 1h(x – 1)

where, ai ≥ 0 and i = 0, 1, 2, 3, … ,(x – 1)

The recursion formulas are the formulas that are used to write the recursive functions or recursive series.

Recursive Formulas

Recursive Formula is a formula that defines each term of sequence using the previous/preceding terms. It defines the following parameters

  • First Term of Sequence
  • Pattern rule to get any term from its previous terms

There are few recursive formulas to find the nth term based on the pattern of the given data. They are,

  • nth term of Arithmetic Progression an = an – 1 + d for n ≥ 2
  • nth term of Geometric Progression an = an – 1 × r for n ≥ 2
  • nth term in fibonacci Sequence an = an – 1 + an – 2 for n ≥ 2 and a0 = 0 & a1 = 1

where

  • d is Common Difference
  • r is Common Ratio

Recursive Formulas For Sequences

Recursive Sequences are the sequences in which the next term of the sequence is dependent on the previous term. One of the most important recursive sequence is the Fibonnaci Sequence, that is represented below as,

0, 1, 1, 2, 3, 5, 8, …

The recursive formulas or the recursion formulas for different kinds of the sequences are,

Recursive Formula for Arithmetic Progression

For Arithmetic Progression the nth term is given using the recursive formula as,

an = a(n-1) + d for n ≥ 2

where,

  • an is the nth term of a A.P.
  • d is the common difference

Recursive Formula for Geometric Progression

For Geometric Progression the nth term is given using the recursive formula as,

an = {a(n-1)}r for n ≥ 2

where,

  • an is the nth term of a G.P.
  • r is the common ratio

Recursive Formula for Fibonacci Sequence

For Fibonacci Sequence the nth term is given using the recursive formula as,

an = a(n-1) + a(n-1) for n ≥ 2

where,

  • a0 = 1
  • a1 = 1
  • an is the nth term of a Fibonacci Sequnece

Useful Sequence And Formulas

Some of the useful sequences and there formulas for the nth term are added in the table below,

Triangular Sequence

1, 3, 6, 10, 15, 21,…

an = n(n+1)/2

Square Sequence

1, 4, 9, 16, 25, 36,…

an = (n)2

Cube Sequence

1, 8, 27, 64, 125, 216, …

an = (n)3

Exponetetial Sequence

2, 4, 8, 16, 32, 64, 128,…

an = 2n

Factorial Sequence

1, 2, 6, 24, 120, 720, 5040,…

an = n!

Read More,

Examples Using Recursive Formula

Example 1: Given a series of numbers with a missing number in middle 1, 11, 21, ?, 41. Using recursive formula find the missing term.

Solution:

Given,

1, 11, 21, …, 41

First term (a) = 1

d = T2 – T1 = T3 – T2

d = 11 – 1 = 21 – 11 = 10

Recursive Function in AP an = an-1 + d

a4 = a4-1 + d

a4 = a3 + d

a4 = 21 + 10

a4 = 31

Example 2: Given series of numbers 5, 9, 13, 17, 21,… From the given series find the recursive formula 

Solution:

Given number series

5, 9, 13, 17, 21,…

First Term (a) = 5

d = T2 – T1 = T3 – T2

d = 9 – 5 = 13 – 9 = 4

Recursive Formula for AP an = an-1 + d

an = an-1 + 4

Example 3: Given a series of numbers with a missing number in middle 1, 3, 9,…,81, 243. Using recursive formula find the missing term.

Solution:

Given,

1, 3, 9,…, 81, 243

First Term (a) = 1

a2/a1 = 3/1 = 3

a3/a2 = 9/3 = 3

a5/a4 = 243/81 = 3

Common Ratio (r) = 3

Recursive Function to find nth term in GP an = an-1 × r

a4 = a4-1 × r

a4 = a3 × r

a4 = 9 × 3

a4 = 27

Example 4: Given series of numbers 2, 4, 8, 16, 32, … From the given series find the recursive formula.

Solution:

Given number series,

2, 4, 8, 16, 32, …

First term (a) = 2

a2/a1 = 4/2 = 2

a3/a2 = 8/4 = 2

a4/a3 = 16/8 = 2

Common Ratio (r) = 2

Recursive Formula an = an-1 × r

an = an-1 × 2

Example 5: Find the 5th term in a Fibonacci series if the 3rd and 4th terms are 2,3 respectively.

Solution:

Given,

  • a3 = 2
  • a4 = 4

Then in Fibonnaci Sequence, a5 = a3 + a4

a5 = 2 + 3

a5 = 5

Practice Question on Recursive Formula

Q1: Find the Recursive Formula for the sequence, 3,7, 11, 15….

Q2: Find the middle term of the sequence, 4, 9, 14, …. 39, 44

Q3: Find the Recursive Formula for the Sequence 44, 40, 36, …..

Q4: Find the middle term of the sequence 6, 9, 12, …. 33

FAQs on Recursive Formula

1. What is Recursive Formula in Math?

Recursive Formula also called the Recursion formula is a formula that give the next term of any sequence depending on the previous terms of the sequence.

2. What is the Recursive Rule For the Fibonacci series?

The recursive formula for the Fibonacci Series is Fn = F(n-1) + F(n-2), where n > 1.

3. What is the Difference Between Recursive and Explicit Formulas?

Recursive Formula is a formula that is used to find the nth term of a series when the previous terms of the sequence are given, where as Explicit Formulas give the nth term of the sequence and is not dependent on the previous terms of the sequence.

4. What is the Recursive Formula for 9, 15, 21, 27?

The recursive formula for the sequence 9, 15, 21, and 27 is, an = an-1 + 6

5. What are Some Recursion Formulas?

Some Famous Recusrion formulas are,

  • Recursive formula of an Arithmetic Sequence is, an = an-1 + d
  • Recursive formula of a Geometric Sequence is, an = (an-1)r


Last Updated : 28 Dec, 2023
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