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Use of induction
Some series are easy to sum by spotting patterns. For example, we have seen that the sum of the first n odd numbers is
1+3+5+⋯+(2n−1)=n2.On the other hand, the formula for the sum of the first n squares,
12+22+32+⋯+n2=16n(n+1)(2n+1),is hardly obvious and requires a proof. While the Greeks had some very creative proofs of results such as this, the best approach is to give a proof using mathematical induction.