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

Question

Using the principle of mathematical induction, prove each of the following for all n ϵ N: 3 n ≥ 2 n .

Philoid · Accepted Answer

To Prove:

Let us prove this question by principle of mathematical induction (PMI) for all natural numbers

Let P(n):

For n = 1 P(n) is true since , which is true

Assume P(k) is true for some positive integer k , ie,

= …(1)

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

Consider ,

[ Using 1 ]

[Multiplying and dividing by 2 on RHS ]

Now ,

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

By the principle of mathematical induction, P(n) is true for all natural numbers, ie, N.

Hence proved.