Question

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

2n + 1 < 2ⁿ, for all natual numbers n ≥ 3.

Answer 1

Let P(n) be the given statement

i.e., P(n) : (2n + 1) < 2n for all natural numbers, n ≥ 3.

We observe that P(3) is true

Since 2.3 + 1 = 7 < 8 = 2³

Assume that P(n) is true for some natural number k

i.e., 2k + 1 < 2^k

To prove P(k + 1) is true

We have to show that 2(k + 1) + 1 < 2^k+1

Now, we have 2(k + 1) + 1 = 2k + 3

= 2k + 1 + 2 < 2^k + 2 < 2^k . 2

= 2^{k + 1}.

Thus P(k + 1) is true, whenever P(k) is true.

Hence, by the Principle of Mathematical Induction P(n) is true for all natural numbers, n ≥ 3.