Mathematical induction is a mathematical technique is used to prove a statement, a theorem or a formula is true for every natural number.
An essential property of the set N = {1, 2, 3, …} of positive integers follows:
Principle of Mathematical Induction I: Let P be a proposition defined on the positive integers N; that is, P(n) is either true or false for each n ∈ N. Suppose P has the following two properties:
(i) P(1) is true.
(ii) P(k + 1) is true whenever P(k) is true.
Then P is true for every positive integer n ∈ N.
We shall not prove this principle. In fact, this principle is usually given as one of the axioms when N is developed axiomatically.
Principle of Mathematical Induction II: Let P be a proposition defined on the positive integers N such that:
(i) P(1) is true.
(ii) P(k) is true whenever P(j) is true for all 1 ≤ j < k.
Then P is true for every positive integer n ∈ N.
Note: Sometimes one wants to prove that a proposition P is true for the set of integers {a, a + 1, a + 2, a + 3, . . .} where a is any integer, possibly zero. This can be done by simply replacing 1 by a in either of the above Principles of Mathematical Induction.
The technique has two steps to prove any statement-
1. Base step (it proves that a statement is true for the initial value)
2. Inductive step (it proves that is the statement is true for n’th iteration then it is also true for (n+1)’th iteration.
Example:
Sol. here for n = 1, 1 = 1²
Now let us assume that the statement is true for n = k
Hence we assume that 
Now we need to prove that-
So that 
Hence-
Example-2:Prove 1+2+…+n=n(n+1)/2 using a proof by induction
Sol.
Let n=1: 1 = 1(1+1)/2 = 1(2)/2 = 1 is true,
Step-1: Assume n=k holds: 1+2+…+k=k(k+1)/2
Show n=k+1 holds: 1+2+…+k+(k+1)=(k+1)((k+1)+1)/2
Substitute k with k+1 in the formula to get these lines. Notice that I write out what I want to prove.
Step-2: Now start with the left side of the equation
1+2+…+(k+1)=1+2+…+k+(k+1)
=k(k+1)/2 + (k+1)
=(k(k+1) + 2(k+1))/2 by 2/2=1 and distribution of division over addition
=(k+2)(k+1)/2 by distribution of multiplication over addition
=(k+1)(k+2)/2 by commutativity of multiplication.
Example-3: Prove the following by using the principle of mathematical induction for all n ∈ N-
Sol. Here, n = 1, 
Step-1: Assume n = k holds-
Now show n = k + 1 also holds-
Consider-
Which is also true for n = k + 1.
Hence proved.