### 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

$1 + 3 + 5 + \dots + (2n-1) = n^2.$

On the other hand, the formula for the sum of the first $$n$$ squares,

$1^2 + 2^2 + 3^2 + \dots + n^2 = \dfrac{1}{6}n(n+1)(2n+1),$

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.

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