Question

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

3ⁿ ≥ 2ⁿ .

Answer 1

To Prove:

3ⁿ ≥ 2ⁿ

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

Let P(n): 3ⁿ ≥ 2ⁿ

For n = 1 P(n) is true since 3ⁿ ≥ 2ⁿ i x e x 3 ≥ 2, which is true

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

= 3^k ≥ 2^k....(1)

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

Consider,

[Multiplying and dividing by 2 on RHS]

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.