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 < 2k
To prove P(k + 1) is true, we have to show that 2(k + 1) + 1 < 2k+1. Now, we have 2(k + 1) + 1 = 2 k + 3
= 2k + 1 + 2 < 2k + 2 < 2k . 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.