Using the principle of mathematical induction, prove each of the following for all n ϵ N:

.

To Prove:

Let us prove this question by principle of mathematical induction (PMI)

Let P(n):

For n = 1

LHS =

RHS =

Hence, LHS = RHS

P(n) is true for n = 1

Assume P(k) is true

= ……(1)

We will prove that P(k + 1) is true

RHS =

LHS =

=

[ Writing the second last term ]

= [ Using 1 ]

=

=

=

( Splitting the numerator and cancelling the common factor)

= RHS

LHS = RHS

Therefore, P (k + 1) is true whenever P(k) is true

By the principle of mathematical induction, P(n) is true for×

where n is a natural number

Hence proved.