Prove statement, by using the Principle of Mathematical Induction for

1.	Prove statement, by using the Principle of Mathematical Induction for all n ∈ N, that :2n + 1 < 2n , for all natural numbers n ≥ 3.
Answer» 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 = 2 k + 3 = 2k + 1 + 2 < 2^k + 2 < 2^k . 2 = 2k + 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.

Prove statement, by using the Principle of Mathematical Induction for all n ∈ N, that :2n + 1 < 2n , for all natural numbers n ≥ 3.

Answer»

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 = 2 k + 3

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