Use mathematical induction to prove that the statement is true for every positive integer n.0.32n < 0.32n-1

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, the left-hand side of the statement is 0.321 = 0.32. The right-hand side becomes 0.321-1 = 0.320 = 1. Since 0.32 < 1, the statement is true for n = 1.
b). Assume the statement is true for n = k:
 0.32k < 0.32k-1
 Multiply both sides by 0.32:
 0.32 ? 0.32k = 0.32k+1 < 0.32 ? 0.32k-1 = 0.32k = 0.32(k+1)-1, or 
  0.32k+1 < 0.32(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.