Use mathematical induction to prove that the statement is true for every positive integer n.2n > 2n - 1

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. then, 21 = 2 > 21 - 1 = 20 = 1. So, the statement is true for n = 1.
b). Assume that the statement is true for n = k:
 2k > 2k - 1
 Multiply both sides by 2:
 2 ? 2k = 2k + 1 > 2 ? 2k - 1 = 2k = 2(k + 1) - 1 or 2k + 1 > 2(k + 1) - 1
Since the statement is true for n = k + 1 as long as it is true for n = k, and since the statement is true for n = 1, then it is true for all natural numbers n.