"Counting in their heads"  1895 oil painting

08112020, 10:39 AM
Post: #15




RE: "Counting in their heads"  1895 oil painting
(08112020 12:33 AM)Gerson W. Barbosa Wrote: It looks like we can proceed to the next level: Sum of cubes = squared triangular number = \(\binom{n+1}{2}^2\) S = (10³+11³+12³+13³+14³) = \(\binom{15}{2}^2  \binom{10}{2}^2\) = (10545)*(105+45) = 60*150 = 9000 → S/450 = 20  We can rewrite formula in terms of central element, c = a + (n1)/2 (x+y)³ + (xy)³ = (x³+3x²y+3xy²+y³) + (x³3x²y+3xy²y³) = 2x³+6xy² sumofncubes \(= c^3 n + 6c×(1^2 + 2^2 + \cdots + ({n1\over2})^2) = c^3 n + 6c × \binom{n+1}{3}/4 \) → sumofncubes = cn*(c² + (n²1)/4) S = 12*5*(12² + (4*6)/4) = 60*150 = 9000 

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