Recursive Function: Definition, Types, Formula & Examples

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Recursive Function is one of the most common functions or expressions used in mathematics. A recursive function is an expression which expresses a series of functions by repetitively using its previous terms.

Recursive Function is based on the principle of arithmetic and geometric sequence.
While calculating the function, two cases are considered, namely the base case and the recursive case.
It is also known as a computable function.
Recursive Function solves a problem while solving smaller cases of similar problems.
It can be written for both arithmetic and geometric functions.
The method is suitable for divide-and-conquer algorithms like merge and sort.
It is used to solve mathematical problems like factorial of a number and generation of fibonacci series.

Read More: Factorial Formula

Table of Content

What is Recursive Function?
Types of Recursive Function
Solved Examples of Recursive Function
Things to Remember
Sample Questions

Key Terms: Recursive Function, Arithmetic Function, Geometric Function, Sequences, Factorial, Fibonacci Series, Algorithms, Java, Python

What is Recursive Function?

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A recursive Function is a function that will call itself directly or indirectly to solve a problem. The word recursive means anything which is reoccurring, that is repeating itself. It is based on the value of more than one variable.

As the name suggests, a recursive function is used to calculate a function's value.
There are two types of recursive function formulas, namely direct recursion and indirect recursion.
For example, you have a measuring cylinder.
Now you, add 1 ml of water to make it 2 ml.
Then, you add another ml of water to it.
Similarly, the entire procedure continues till the desired volume.
Thus, here, the base value, 1ml, kept recurring.

Types of Recursive Function

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The two types of recursive function formulas are as follows:

Arithmetic Functions

Arithmetic functions are characterized by a common difference between the subsequent terms. The recursive formula for an arithmetic function having sequence f₁, f₂, f₃, ….f_n can be written as:

f_n = f_n-1 + d

How to calculate the recursive function?

The process to calculate the recursive functions using the arithmetic functions are as follows:

First determine the common difference d.
State the first term.
Create the formula of the previous term + common difference.

For Geometric Functions

Geometric functions are characterized by a common ratio between the subsequent terms. The recursive formula for a geometric function can be written as:

f_n= r. f_n-1

How to calculate the recursive function?

The process to calculate the recursive functions using the geometric functions are as follows:

First determine the common ratio r.
Next, state the first term.
Then create the formula to be the previous term x common ratio.

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Solved Examples of Recursive Functions

Given below are some examples of recursive functions.

Example 1. Let a₁= 5 and a_n= 2a_n-1+ 1. Determine the value of a₄.

Ans. Given a₁ = 5

Next determine the value of a₂by putting n = 2 in the equation a_n= 2a_n-1+ 1.
When we put n = 2 then equation will become: a₂ = 2a₁+ 1 = 2 x 5 + 1 = 11
So value of a₂ is 11.
Next by using similar method calculate the value of a₃ and a₄.
For a₃ and a_4,put n = 3 and 4
Then equation for a₃= 2a₂+ 1 = 2 x 11 + 1 = 23
Equation for a₄ = 2a₃+ 1 = 2 x 23 + 1= 47

Example 2. Let a₁= 2 and a_n = 5. a_n-1. Determine the value of a₄.

Ans. Given a₁ = 2

Next determine the value of a₂by putting n = 2 in the equation a_n = 5. a_n-1.
When we put n = 2 then equation will become: a₂ = 5 x a₁= 2 x 5 = 10
So value of a₂ is 10.
Next by using similar method calculate the value of a₃ and a₄.
For a₃ and a_4,put n = 3 and 4
Then equation for a₃= 5 x a₂= 2 x 10 = 20
Equation for a₄ = 5 x a₃= 2 x 20 = 40

Read More: Complex Numbers and Quadratic Equations

Example 3. Let a₁= 10 and a_n= a_n-1+ 10. Determine the value of a₅.

Ans. Given a₁ = 10

Next determine the value of a₂by putting n = 2 in the equation a_n= a_n-1+ 10.
When we put n = 2 then equation will become: a₂ = a₁+ 10 = 10 + 10 = 20
So value of a₂ is 20.
Next by using similar method calculate the value of a₃ and a₄.
For a₃ , a₄, and a₅ put n = 3, 4 and 5
Then equation for a₃= a₂+ 10 = 20 + 10 = 30
Equation for a₄ = a₃+ 10 = 30 + 10= 40
Equation for a₅= a₄+ 10 = 40 + 10= 50

Example 4. Let a₀= 1 and a_n = 5. na_n-1. Determine the value of a₃.

Ans. Given a₀= 1

Next determine the value of a₁by putting n = 1 in the equation a_n = 5. na_n-1.
When we put n = 1 then equation will become: a₁ = 5 x na₀= 5 x 1 x 1 = 5
So value of a₁ is 5.
Next by using similar method calculate the value of a2 and a3.
For a₂ and a_3,put n = 2 and 3
Then equation for a₂= 5 x na₁= 5 x 5 x 2 = 50
Equation for a₃ = 5 x na₂= 5 x 3 x 50 = 750

Read More: Types of Relation

Things to Remember

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A recursive function is an expression which expresses a series of functions by repetitively using its previous terms.
It can be written for both arithmetic and geometric functions.
A recursive function is highly used in mathematics.
It is also used in computer programming languages, such as C++, Java, Python, etc.
The two parts of a recursive function are: the first term of sequence and the pattern or the rule followed by the sequence.
Arithmetic functions are characterized by a common difference between the subsequent terms.
The recursive formula for an arithmetic function is f_n = f_n-1 + d.
Geometric functions are characterized by a common ratio between the subsequent terms.
The recursive formula for a geometric function is f_n= r. f_n-1

Sample Questions

Ques. Find the recursive formula for the sequences 3, 6, 9, 12, 15, 18, 21. (3 Marks)

Ans. Given sequence, 3, 6, 9, 12, 15, 18, 21.

The given sequence is a geometric sequence because if the preceding term is multiplied by 3, we get the successive terms.
Thus, r = 3
Hence, the recursive formula is:
f(1) = 3 , f(n)= 3. f(n-1)

Ques. Find the recursive formula for the sequence 3, 4, 5, 6, 7, 8. (3 Marks)

Ans. Given sequence, 3, 4, 5, 6, 7, 8.

The given sequence is an arithmetic sequence because if the preceding term is lesser than by 1, we get the successive terms.
Thus, d = 1
Hence, the recursive formula is:
f(1) = 3 , f(n)= 1 + f(n-1)

Ques. Find the recursive formula for the sequence 2, 4, 8, 16, 32. (3 Marks)

Ans. Given sequence, 2, 4, 8, 16, 32.

The given sequence is a geometric sequence because if the preceding term is multiplied by 2, we get the successive terms.
Thus, r = 2
Hence, the recursive formula is:
f(1) = 2 , f(n)= 3. f(n-1)

Ques. Find the recursive formula for the sequence 3, 7, 11, 15, 19 (3 Marks)

Ans. Given sequence, 3, 7, 11, 15, 19

The given sequence is an arithmetic sequence because if the preceding term is lesser than by 4, we get the successive terms.
Thus, d = 4
Hence, the recursive formula is:
f(1) = 3 , f(n)= 4 + f(n-1)

Ques. Find the recursive formula for the sequence 6, 12, 24, 48. (3 Marks)

Ans. Given sequence, 6, 12, 24, 48..

The given sequence is a geometric sequence because if the preceding term is multiplied by 2, we get the successive terms.
Thus, r = 2
Hence, the recursive formula is:
f(1) = 6 , f(n)= 2. f(n-1)

Ques. Find the recursive formula for the sequence 9, 19, 29, 39, 49. (3 Marks)

Ans. Given sequence, 9, 19, 29, 39, 49..

The given sequence is an arithmetic sequence because if the preceding term is less than by 10, we get the successive terms.
Thus, d = 10
Hence, the recursive formula is:
f(1) = 9 , f(n)= 10 + f(n-1)

Ques. Find the recursive formula for the sequence 4, 16, 64, 96 (3 Marks)

Ans. Given sequence, 4, 16, 64, 96

The given sequence is a geometric sequence because if the preceding term is multiplied by 4, we get the successive terms.
Thus, r = 4
Hence, the recursive formula is:
f(1) = 4 , f(n)= 4. f(n-1)

Ques. Find the recursive formula for the sequence 2, 4, 6, 8, 10, 12. (3 Marks)

Ans. Given sequence, 2, 4, 6, 8, 10, 12.

The given sequence is an arithmetic sequence because if the preceding term is less than by 2, we get the successive terms.
Thus, d = 2
Hence, the recursive formula is:
f(1) = 2 , f(n)= 2 + f(n-1)

Ques. Let a₁= 4 and a_n= 2a_n-1+ 2. Determine the value of a_5. (3 Marks)

Ans. Given a₁ = 2

Next determine the value of a₂by putting n = 2 in the equation a_n= 2a_n-1+ 2.
When we put n = 2 then equation will become: a₂ = 2a₁+ 1 = 2 x 4 + 2 = 10
So value of a₂ is 10.
Next by using similar method calculate the value of a₃ and a₄.
For a₃ and a_4,put n = 3 and 4
Then equation for a₃= 2a₂+ 2 = 2 x 10 + 2 = 22
Equation for a₄ = 2a₃+ 1 = 2 x 22 + 2= 46
Equation for a₅ = 2a₄ + 1 = 2 x 46 + 2= 94

Ques. Let a₁= 5 and a_n = 10 a_n-1. Determine the value of a₄. (3 Marks)

Ans. Given a₁ = 5

Next determine the value of a₂by putting n = 2 in the equation a_n = 10 . a_n-1.
When we put n = 2 then equation will become: a₂ = 10 x a₁= 5 x 10 = 50
So value of a₂ is 50.
Next by using similar method calculate the value of a₃ and a₄.
For a₃ and a_4,put n = 3 and 4
Then equation for a₃= 10 x a₂= 10 x 50 = 500
Equation for a₄ = 10 x a₃= 10 x 500 = 5000

Ques. Let f(0) = 0 for n greater than equal to 1, f(n) = f(n-1) + 5. Determine the value of f(1), f(2) and f(3). (3 Marks)

Ans. Given f(0) = 0

Next determine the value f(1) by putting n = 1 in the equation f(n) = f(n-1) + 5.
When we put n = 1 then equation will become: f(1) = f(0) + 5 = 0 + 5 = 5.
So value of f(1) is 5.
For f(2) and f(3), put n = 2 and 3 in the similar equation
Then equation for f(2) = f(1) + 5 = 5 + 5 = 10.
Equation for f(3) = f(2) + 5 = 10 + 5 = 15.

Ques. Let a₁= 20 and a_n = 2. a_n-1. Determine the value of a₄. (3 Marks)

Ans. Given a₁ = 20

Next determine the value of a₂by putting n = 2 in the equation a_n = 2 . a_n-1.
When we put n = 2 then equation will become: a₂ = 10 x a₁= 2 x 20 = 40
So value of a₂ is 40.
Next by using similar method calculate the value of a₃ and a₄.
For a₃ and a_4,put n = 3 and 4
Then equation for a₃= 2 x a₂= 2 x 40 = 80
Equation for a₄ = 2 x a₃= 2 x 80 = 160

Ques. Let f(0) = 1 for n greater than equal to 1, f(n) = n!. Determine the value of f(1), f(2) and f(3). (3 Marks)

Ans. Given f(0) = 1

Next determine the value f(1) by putting n = 1 in the equation f(n) = n!.
When we put n = 1 then equation will become: f(1) = 1 x 1 = 1.
So value of f(1) is 1.
For f(2) and f(3), put n = 2 and 3 in the similar equation
Then equation for f(2) = 1 x 2 = 2.
Equation for f(3) = 1 x 2 x 3 = 6.

Ques. Let a_n = 5ⁿ_. Determine the value of a₄. (3 Marks)

Ans. Given a_n = 5ⁿ

Next determine the value of a1by putting n = 1 in the equation a_n = 5ⁿ.
When we put n = 1 then equation will become: a1 = 5¹= 5
So value of a₁is 5.
Next by using similar method calculate the value of a₂, a₃ and a₄.
For a_2, a₃ and a_4,put n = 2, 3 and 4
Equation for a₂ = 5² = 25
Then equation for a₃= 5³= 125
Equation for a₄ = 5⁴= 625

Ques. Let f(0) = 1 for n greater than equal to 1, f(n) = 6n – 1. Determine the value of f(1), f(2) and f(3). (3 Marks)

Ans. Given f(0) = 1

Next determine the value f(1) by putting n = 1 in the equation f(n) = 6n -1.
When we put n = 1 then equation will become: f(1) = 6 x 1 – 1= 5.
So value of f(1) is 5.
For f(2) and f(3), put n = 2 and 3 in the similar equation
Then equation for f(2) = 6 x 2 – 1 = 11
Equation for f(3) = 6 x 3 – 1 = 17

Ques. Give the recursive definition with initial condition for the function f(n) = 3n + 2, determine the value of n = 1,2,3,4. (3 Marks)

Ans. Given f(n) = 3n + 2

Next determine the value f(1) by putting n = 1 in the equation f(n) = 3n + 2.
When we put n = 1 then equation will become: f(1) = 3 x 1 + 2= 5.
So value of f(1) is 5.
For f(2) and f(3), put n = 2 and 3 in the similar equation
Then equation for f(2) = 3 x 2 + 1 = 7
Equation for f(3) = 3 x 3 + 2 = 11
Equation for f(4) = 3 x 4 + 2 = 14

Ques. The 13^th and 14^th terms of the Fibonacci sequence are 14 and 20 respectively. Find the 15^th term. (2 Marks)

Ans. Using the recursive formula for the 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

= 14 + 20

= 34

Ques. Assume a recursive function on positive integers where f(0) = 2 and function is given as f(n+1) = 2f(n) + 1. Determine the value of n = 1,2,3. (3 Marks)

Ans. Given f(n+1) = 2f(n) + 3

Next determine the value f(1) by putting n = 0 in the equation f(n+1) = 2f(n) + 3
When we put n = 1 then equation will become: f(1) = 2 x 2 + 1 = 5.
So value of f(1) is 5.
For f(2) and f(3), put n = 1 and 2 in the similar equation
Then equation for f(2) = 5 x 2 + 1 = 11
Equation for f(3) = 11 x 3 + 1 = 34

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Recursive Function: Definition, Types, Formula & Examples

What is Recursive Function?

Types of Recursive Function

Arithmetic Functions

How to calculate the recursive function?

For Geometric Functions

How to calculate the recursive function?

Solved Examples of Recursive Functions

Things to Remember

Sample Questions

Comments

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