Use mathematical induction to prove the following.0.9n < 0.9n-1
What will be an ideal response?
Answers may vary. One possibility:
Sn: 0.9n < 0.9n-1
S1: 0.91 < 0.91-1
Sk: 0.9k < 0.9k-1
Sk+1: 0.9k+1 < 0.9k
1. Basis step: Since 0.91-1 = 0.90 = 1 and 0.91 < 1, S1 is true.
2. Induction step: Let k be any natural number. Assume Sk. Deduce Sk+1.
0.9k < 0.9k-1
0.9k ? 0.9 < 0.9k-1 ? 0.9 Multiplying by 0.9, 0.9 > 0
0.9k+1 < 0.9(k-1)+1
0.9k+1 < 0.9k.
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