Use mathematical induction to prove that the statement is true for every positive integer n.0.23n < 0.23n-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 0.681 = 0.68. The right-hand side becomes 0.681-1 = 0.680 = 1. Since 0.68 < 1, the statement is true for n = 1.
b). Assume the statement is true for n = k:
 0.23k < 0.23k-1
 Multiply both sides by 0.68:
 0.68?0.23k = 0.68k+1 < 0.68?0.23k-1 = 0.68k = 0.68(k+1)-1, or 
  0.68k+1 < 0.68(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.