Use mathematical induction to prove that the statement is true for every positive integer n.8n > n

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, 81 = 8 > 1, so the statement is true for n = 1.
b). Assume that the statement is true for n = k:
 8k > k
 Multiply both sides by 8:
 8 ? 8k = 8k + 1 > 8k
 Also, since k ? 1, then 8k > 1, which can be re-written as 8k > k + 1. Thus,
  8k + 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.