Prove the following by the principle of mathematical induction: i.e., the sum of the first n natural numbers is

Question

Philoid · Accepted Answer

Let us Assume P(n) = 1 + 2 + 3 + - - - - - - + n =

For n = 1

L.H.S of P(n) = 1

R.H.S of P(n) = = 1

Therefore, L.H.S =R.H.S

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

Let assume P(n) be the true for n = k, so

1 + 2 + 3 + - - - - - + k = - - - (1)

Now

(1 + 2 + 3 + - - + k) + (k + 1)

= + (k + 1)

= (k + 1)

=

P(n) is true for n = k + 1

P(n) is true for all n∈N

So , by the principle of Mathematical Induction

Hence, P(n) = 1 + 2 + 3 + - - - + n = is true for all n∈N