1. Use induction to prove that
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 ,
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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