Using the principle of mathematical induction, prove each of the following for all n ϵ N:
= (n + 1).

Question

Using the principle of mathematical induction, prove each of the following for all n ϵ N: = (n + 1).

Philoid · Accepted Answer

To Prove:

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

Let P(n):

For n = 1

LHS =

RHS = = 2

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 =

[Now writing the second last term ]

=

= [ Using 1 ]

=

= k + 2

= 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.