Use mathematical induction to prove that the statement is true for every positive integer n.0.63n < 0.63n-1
What will be an ideal response?
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:
0.63k < 0.63k-1
Multiply both sides by 0.68:
0.68?0.63k = 0.68k+1 < 0.68?0.63k-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.
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What will be an ideal response?