# Solutions to Problem Set 13

Math 310-01/02

10-13-2017

1. Use induction to prove that

For ,

The result holds for .

Assume that and that the result holds for n:

I must prove the result for . Using the induction hypothesis, I have

This proves the result for . Hence, the result is true for all , by induction.

2. A sequence of integers is defined by

Prove that for ,

For ,

For ,

Assume the result is true for all . In particular, it is true for and for . So

This proves the result for n. Hence, the result is true for all by induction.

Our experience is composed rather of illusions lost than of wisdom acquired. - Joseph Roux

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