The number of squares in a n × n chess board will be ∑ n2; n varying from 1 to n.
Now, ∑ n2 = 12 + 22 + 32 + ⋯ n2
= \(\frac{n(n\,+\,1)(2n\,+\,1)}{6}\) …. (i)
Put n = 8 in eq (i), we get
∑ 82 = \(\frac{8\,\times\, (8\, +\, 1)(2\, ×\, 8 ×\, 1)}{6}\) = \(\frac{8 \,×\, 9\, ×\, 17}{6}\) = 204