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…… an, 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 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:
|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.|
It has two components:
It has two steps:
|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.|