How do you prove by induction?

First you prove the case n=1 is true. Then you assume that n=k is true, then calculate what n=k+1 is. This should prove true, so that by induction, you have proved that the statement is true for all natural numbers.

PK

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I do not understand this topic and particularly this example. In the class the result was found out but I still do not get it. How did the teacher came up with this outcome?