Here are the problems. They will teach you how to prove mathematical statements by using induction method.
Problem 1
Let such that , for a fixed integer . Prove that positive, .
Proof
For , (true)
Let , we need to show that
So,
Suppose it is true for , i.e., , we show that it is also true for
Thus, , hence shown.
Problem 2
Let be a group and suppose that . Show that for any positive integer
Proof
The statement for is simply which is certainly true.
For
Hence the result is true for
Now, assume that the result holds for
Using this induction hypothesis, we have
The statement holds for so by induction it holds for all values of
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