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1.19 The sum of squares

Example

To prove that

1^{2}+2^{2}+3^{2}+\dots+n^{2}=6^{-1}n(n+1)(2n+1)

holds for all positive integers $n\,$ .

Proof by induction. Let $P(n)\,$ be the statement

P(n):\quad 1^{2}+2^{2}+3^{2}+\dots+n^{2}=6^{-1}n(n+1)(2n+1).

(i) Basis of induction: $P(1)\,$ asserts that $1^{2}=6^{-1}\times 1\times 2\times 3\,$ , which is true.