Use mathematical induction to prove that the statement is true for every positive integer n.5n > 1

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, 51 = 5 > 1, so the statement is true for n = 1.
b). Assume the statement is true for n = k:
 5k > 1
 Multiply both sides by 9:
 5 ? 5k = 5k + 1 > 1 ? 5 = 5 or 5k + 1 > 5. Also, 5 > 1, so 5k + 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.