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Let be the proposition that for the positive integer
(a) is the statement that
(b) Basic Step
is true because
This completes the basis step.
(c) For the inductive hypothesis, we assume that is true for an arbitrary
positive integer That is, we assume that
(d) Inductive Step
To carry out the inductive step we must show that when we assume that is true, then is also true. That is, we must show that
assuming the inductive hypothesis
(e) Under the assumption of we see that
Note that we used the inductive hypothesis in the second equation in this chain of equalities to replace with
We have completed the inductive step.
(f) Because we have completed the basis step and the inductive step, by mathematical induction we know that is true for all positive integers That is, for the positive integer
5. Let be the proposition that