"Counting in their heads"  1895 oil painting

08102020, 12:49 PM
(This post was last modified: 08102020 02:38 PM by Albert Chan.)
Post: #9




RE: "Counting in their heads"  1895 oil painting
(08102020 11:50 AM)Gerson W. Barbosa Wrote: \( \(s(x) = (a+bx)^2 = a^2 + 2abx + b^2x^2 = a² + b(2a+b)\binom{x}{1} + 2b^2\binom{x}{2}\) → Σ(s(x), x=0..n1) = [a², b(2a+b), 2b²] • [nC1, nC2, nC3] With a=55, b=10, n=10 S = [a², b(2a+b), 2b²] • [nC1, nC2, nC3] = [3025, 1200, 200] • [10, 45, 120] = 30250 + 54000 + 24000 = 108250 → S / 10825 = 10  We can also use central element, c = a + b(n1)/2 → S = c²n + b²(n+1)(n)(n1)/12 With c = (55+145)/2 = 100, b=10, n=10: S = 100²*10 + 10²*11*10*9/12 = 100000 + 8250 = 108250 Or, simply force b=1 (55² + 65² + ... + 145²) / 10825 = (5.5² + 6.5² + ... + 14.5²) / 108.25 // c=n=10 = (1000 + 11*10*9/12) / 108.25 = 10 

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