Recursive Function in Maths

Mathematics often deals with patterns and sequences. One powerful way to define such sequences is through recursive functions. A recursive function is a function that refers to itself in its definition. This concept is widely used in sequences, series, and even in computer algorithms.

It usually has two main components:

• Base Case: The simplest instance of the problem, which has a direct answer.
• Recursive Case: Defines the function in terms of itself for smaller inputs.

Mathematically, a general recursive function can be expressed as:

h(x) = a₀h(0) + a₁h(1) + a₂h(2) + ... + a_{x - 1}h(x - 1)

where,

• a_i ≥ 0
• 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 Formula

The recursion formula is the formula used to write recursive functions or recursive series. It provides a way to generate the sequence step by step using previous terms.

• Base case: Gives the first term(s) of the sequence.
• Recursive formula: Expresses the xxx-th term using one or more previous terms.

Recursive Formulas For Various Sequences

Recursive Sequences are sequences in which the next term of the sequence is dependent on the previous term. One of the most important recursive sequences is the Fibonacci Sequence:

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

Solved Question

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 the 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 Fibonacci Sequence, a₅ = a₃ + a₄

a₅ = 2 + 3
⇒ a₅ = 5

Unsolved Question on Recursive Function

Question 1: Define a sequence recursively as: a₁ = 2, a_n+1 = 3a_n + 1 for n ≥ 1. Find a₃.

Question 2: A sequence is defined recursively as: b₁ = 1, b₂ = 1, b_n = b_n−1 + b_n−2 for n ≥ 3 Find b₅​.

Question 3: A function is defined recursively as: g(0) = 0, g(n) = g(n − 1) + 2n − 1 for n ≥ 1. Find g(4).

Question 2: A sequence is defined recursively as: b₁ = 1, b₂ = 1, b_n = b_n−1 + b_n−2 for n ≥ 3. Find b₅​.

Question 4: A function is defined recursively as: h(0) = 1, h(n) = n⋅h(n − 1) for n ≥ 1. Find h(4).