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"Counting in their heads" - 1895 oil painting
08-09-2020, 02:24 PM
Post: #6
RE: "Counting in their heads" - 1895 oil painting
An elegant way to get sum of squares formula is with geometry.
[Image: 93d0fee733e2156bbedcb4a2854d1483.png]

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
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RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-09-2020 02:24 PM



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