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

Question

Philoid · Accepted Answer

To Prove:

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 Last second term ]

=

= [ Using 1 ]

=

= ( Taking LCM and simplifying )

=

= RHS

Therefore, =

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

Put k = n - 1

=

Hence proved.