k^2 - (1/6) k (k + 1) (2k + 1)
= - (1/6) k (- 6k + (k + 1) (2k + 1))
= - (1/6) k (2 k^2 - 3k + 1)
= - (1/6) k (k - 1) (2k - 1)
= - (1/6) (k-1) k (2k - 1)
= - (1/6) (k-1) ((k-1) + 1) (2(k-1) + 1)
1^2 + 2^2 + ‥‥ + (n-2)^2 + (n-1)^2 + n^2 - (1/6) n (n + 1) (2n + 1)
= 1^2 + 2^2 + ‥‥ + (n-2)^2 + (n-1)^2 - (1/6)(n-1)((n-1) + 1)(2(n-1) + 1)
= 1^2 + 2^2 + ‥‥ + (n-2)^2 - (1/6)(n-2)((n-2) + 1)(2(n-2) + 1)
‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥
‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥
= 1^2 + 2^2 - (1/6)2(2 + 1)(2・2 + 1)
= 1^2 - (1/6)1(1 + 1)(2・1 + 1) = 1 - 1 = 0
よって、
1^2 + 2^2 + ‥‥ + (n-2)^2 + (n-1)^2 + n^2 = (1/6) n (n + 1) (2n + 1)
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