Question

Prove that, 2n+1 < 2ⁿ , for all natural number 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(n) 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 number n ≥ 3.