Step1: Let P(n) = 1 + 2 + 3 + ..............+ n = n(n + 1)/2

Step2: Let n = 1

LHS = 1

RHS = n(n + 1)/2 = 1(1 + 1)/2 = 2/2 = 1

Since LHS = RHS

So, P(n) is true for n = 1

Step3: Assume P(K) is to be true then prove P(k + 1) is true.

Let P(k) = 1 + 2 + 3 + ..............+ k = k(k + 1)/2 is true.   ...............1

Now, We have to prove that P(K + 1) is true.

1 + 2 + 3 + ..............+ (k + 1) = (k + 1)(k + 2)/2

1 + 2 + 3 + .............. k + (k + 1) = (k + 1)(k + 2)/2 .............2

From equation 1, add k + 1 on both side, we get

1 + 2 + 3 + ..............+ k + (k + 1) = k(k + 1)/2 + (k + 1)

1 + 2 + 3 + ..............+ k + (k + 1) = {k(k + 1) + 2(k + 1)}/2

1 + 2 + 3 + ..............+ k + (k + 1) = (k + 1) * (k + 2)/2

which is same as P(K + 1)

So, P(k + 1) is true when P(K) is true.

So, by the principle of mathematical induction, P(n) is true for all n, where n is a natural number.