Recursive Formula

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) + … + a_{x – 1}h(x – 1)

where, a_i ≥ 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 n^th term based on the pattern of the given data. They are,

• n^th term of Arithmetic Progression a_n = a_{n – 1} + d for n ≥ 2
• n^th term of Geometric Progression a_n = a_{n – 1} × r for n ≥ 2
• n^th term in fibonacci Sequence a_n = a_{n – 1} + a_{n – 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 n^th term is given using the recursive formula as,

a_n = a_(n-1) + d for n ≥ 2

where,

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

Recursive Formula for Geometric Progression

For Geometric Progression the n^th term is given using the recursive formula as,

a_n = {a_(n-1)}r for n ≥ 2

where,

• a_n is the n^th term of a G.P.
• r is the common ratio

Recursive Formula for Fibonacci Sequence

For Fibonacci Sequence the n^th term is given using the recursive formula as,

a_n = a_(n-1) + a_(n-1) for n ≥ 2

where,

• a₀ = 1
• a₁ = 1
• a_n is the n^th term of a Fibonacci Sequnece

Useful Sequence And Formulas

Some of the useful sequences and there formulas for the n^th 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,

• Golden Ratio
• Harmonic Progression

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 a_n = a_n-1 + d

a₄ = a_4-1 + 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 a_n = a_n-1 + d

a_n = a_n-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 n^th term in GP a_n = a_n-1 × r

a₄ = a_4-1 × 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 a_n = a_n-1 × r

a_n = a_n-1 × 2

Example 5: Find the 5^th term in a Fibonacci series if the 3^rd and 4^th 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

Recursive Formula – FAQs

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 F_n = 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, a_n = a_n-1 + 6

5. What are Some Recursion Formulas?

Some Famous Recusrion formulas are,

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