Use mathematical induction to prove the following.0.25n < 0.25n-1

Question

What will be an ideal response?

Answer 1

Answers may vary. One possibility:
Sn: 0.25n < 0.25n-1
S1: 0.251 < 0.251-1
Sk: 0.25k < 0.25k-1
Sk+1: 0.25k+1 < 0.25k
1. Basis step: Since 0.251-1 = 0.250 = 1 and 0.251 < 1, S1 is true.
2. Induction step: Let k be any natural number. Assume Sk. Deduce Sk+1.
0.25k < 0.25k-1
0.25k ? 0.25 < 0.25k-1 ? 0.25 Multiplying by 0.25, 0.25 > 0
0.25k+1 < 0.25(k-1)+1
0.25k+1 < 0.25k.