Use mathematical induction to prove that the statement is true for every positive integer n.If 0 < a < 1, then an < an-1.(Assume that a is a constant.)

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, a1 < a1-1 = 1, or a < 1, which is true by assumption. Thus, the statement is true for n = 1.
b). Assume the statement is true for n = k:
  ak < ak-1
 Multiply both sides by a:
 a?ak < ak-1?a
 Using the product rule for exponents:
 ak+1 < ak = a(k+1) - 1
The statement is true for n = k + 1 as long as it is true for n = k. Futhermore, the statement is true for n = 1. Thus, the statement is true for all natural numbers n.