Principle of mathematical induction to prove 1+2+2²+2³+....+2^n-1=2^n-1
Let be the proposition that for the first positive integers
Basis Step:
is true, because
Inductive Step:
For the inductive hypothesis we assume that holds for an arbitrary
positive integer That is, we assume that
Under this assumption, it must be shown that is true, namely, that
is also true.
This last equation shows that is true under the assumption that is true. This completes the inductive step.
We have completed the basis step and the inductive step, so by mathematical induction we know that is true for all positive integers That is, we have proved that
for all positive integers
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