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

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, 31 = 3 > 1, so the statement is true for n = 1.
b). Assume that the statement is true for n = k:
 3k > k
 Multiply both sides by 3:
 3?3k = 3k+1 > 3k
 Also, since k ? 1, then 2k > 1, which can be re-written as 3k > k + 1. Thus,
  3k+1 > k + 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.