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We have the arithmetic progression with the first term and the common difference
ii) Let be the proposition that the sum of the first odd positive integers, is
is true, because
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.
When we add to both sides of the equation in we obtain
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 n. That is, we have proven that the sum of the first odd numbers for all positive integres .