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

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.

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