Question

Prove by using principle of mathematical induction : 2ⁿ < 3ⁿ , n ∈ N 

Answer 1

For n = 1, 2¹ < 3¹ which is true. 

Let us assume that the result is true for n = k i.e. 2^k < 3^k . 

Now 2^k < 3^k ⇒ 2(2k^k) < 2(3^k) < 3(3^k) ⇒ 2^k+1 < 3^k+1 

⇒ true for n = k + 1 

Hence by PMI 2ⁿ < 3ⁿ for all n ∈ N