Use mathematical induction to prove the following.If a is a constant and 0 < a < 1, then an < 1.

What will be an ideal response?

Answers may vary. One possibility:
Sn: If a is a constant and 0 < a < 1, then an < 1.
S1: If a is a constant and 0 < a < 1, then a1 < 1.
Sk: If a is a constant and 0 < a < 1, then ak < 1.
Sk+1: If a is a constant and 0 < a < 1, then ak+1 < 1.
1. Basis step: Since it is given that a < 1, then a1 < 1. Therefore, S1 is true.
2. Induction step: Let k be any natural number. Assume Sk. Deduce Sk+1.
If a is a constant and 0 < a < 1, then ak < 1.
If a is a constant and 0 < a < 1, then ak ? a < 1 ? a. Multiplying by a, a > 0
If a is a constant and 0 < a < 1, then ak+1  < a.
If a is a constant and 0 < a < 1, then ak+1  < a < 1. Given a < 1
If a is a constant and 0 < a < 1, then ak+1  < 1.

Mathematics

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