Difference Between Recursion and Induction

Recursion and induction belong to the branch of Mathematics, these terms are used interchangeably. But there are some differences between these terms. 

Recursion is a process in which a function gets repeated again and again until some base function is satisfied. It repeats and uses its previous values to form a sequence. The procedure applies a certain relation to the given function again and again until some base condition is met. It consists of two components: 

1) Base condition: In order to stop a recursive function, a condition is needed. This is known as a base condition. Base condition is very important. If the base condition is missing from the code then the function can enter into an infinite loop. 

2) Recursive step: It divides a big problem into small instances that are solved by the recursive function and later on recombined in the results.

Let a₁, a₂…… a_n, be a sequence. The recursive formula is given by:

an = an-1 + a1

Example: The definition of the Fibonacci series is a recursive one. It is often given by the relation:

F N = FN-1 + FN-2   
where F0 = 0

How to perform Recursion? 

Suppose the function given is

Tn = Tn-1  + C

We first use the given base condition. Let us denote base condition by T₀, n= 2. we will find T₁. C is the constant. 

Tn = Tn-1  + C //n=2
T2  = T2-1 + C
T1 = T1-1 + C
    = T0 + C + C // Base condition achieved, recursion terminates.

Induction

Induction is the branch of mathematics that is used to prove a result, or a formula, or a statement, or a theorem. It is used to establish the validity of a theorem or result. It has two working rules:

1) Base Step: It helps us to prove that the given statement is true for some initial value. 

2) Inductive Step: It states that if the theorem is true for the nth term, then the statement is true for (n+1)th term. 

Example: The assertion is that the nth Fibonacci number is at most 2ⁿ. 

How to Prove a statement using induction? 

Step 1: Prove or verify that the statement is true for n=1

Step 2: Assume that the statement is true for n=k

Step 3: Verify that the statement is true for n=k+1, then it can be concluded that the statement is true for n. 

Difference between Recursion and Induction:

S.No.	Recursion	Induction
1.	Recursion is the process in which a function is called again and again until some base condition is met.	Induction is the way of proving a mathematical statement.
2.	It is the way of defining in a repetitive manner.	It is the way of proving.
3.	It starts from nth term till the base case.	It starts from the initial till (n+1)th term.
4.	It has two components: Base condition Recursive step.	It has two steps: Base step Inductive step
5.	We backtrack at each step to replace the previous values with the answers using the function.	We just prove that the statement is true for n=1. Then we assume that n = k is true. Then we prove for n=k+1.
6.	No assumptions are made.	The assumption is made for n= k
7.	Recursive function is always called to find successive terms.	Here statements or theorems are proved and no terms are found.
8.	It can lead to infinity if no base condition is given.	There is no concept of infinity.