"Counting in their heads" - 1895 oil painting

[Albert Chan]

An elegant way to get sum of squares formula is with geometry.


sum-of-n-squares = 1/3 * (2*n+1) * n*(n+1)/2 = n*(n+1)*(2*n+1)/6

Falling factorial derivation is also simple:

s(x) = x² = x + x*(x-1) = x¹ + x²
S(x) = Σ(s(t), t=0 .. x-1) = x²/2 + x³/3
       = x*(x-1)*(1/2 + (x-2)/3)
       = x*(x-1)*(2*x-1)/6                     // note: sum-of-n-squares = S(n+1)

10²+11²+12²+13²+14²
= S(15) - S(10)
= 15*14*(30-1)/6 - 10*9*(20-1)/6
= 35*(30-1) - 15*(20-1)
= (1050-300) - (35-15)
= 730