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

Question

What will be an ideal response?

Answer 1

Answers may vary. One possibility:
Sn: If a is a constant and 0 < a < 1, then an < an-1.
S1: If a is a constant and 0 < a < 1, then a1 < a1-1.
Sk: If a is a constant and 0 < a < 1, then ak < ak-1.
Sk+1: If a is a constant and 0 < a < 1, then ak+1 < ak.
1. Basis step: Since a1-1 = a0 = 1 and it is given that a < 1 (which implies a1 < 1), 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 < ak-1.
If a is a constant and 0 < a < 1, then ak ? a < ak-1 ? a. Multiplying by a, a > 0
If a is a constant and 0 < a < 1, then ak+1  < a(k-1)+1.
If a is a constant and 0 < a < 1, then ak+1  < ak.