Use mathematical induction to prove that the statement is true for every positive integer n.9n > 9n - 1

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. then, 91 = 9 > 91 - 1 = 90 = 1. So, the statement is true for n = 1.
b). Assume that the statement is true for n = k:
 9k > 9k - 1
 Multiply both sides by 9:
 9 ? 9k = 9k + 1 > 9 ? 9k - 1 = 9k = 9(k + 1) - 1 or 9k + 1 > 9(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.