"Counting in their heads" - 1895 oil painting

[Albert Chan]

(08-08-2020 05:29 PM)EdS2 Wrote:  calculate (10²+11²+12²+13²+14²)/365

With only 5 terms, it is trivial to calculate.
(x-y)² + (x+y)² = 2(x²+y²)

12² + (11²+13²) + (10²+14²) = 5*12² + 2*(1+4) = 730

730/365 = 2

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Here is a way to fit it to polynomial, then "integrate" (see Funny Factorial and Slick Sums)
We know f(x) is quadratic, thus only 3 points needed.

Code:

x    f(x)   Δ     Δ² 
0    10²
1    11²    21
2    12²    23    2

\(f(x) = 100 + 21\binom{x}{1} + 2\binom{x}{2}\)

\(F(x) = \sum_{t=0}^{x-1} f(t) = 100\binom{x}{1} + 21\binom{x}{2} + 2\binom{x}{3}\)

F(5)/365 = (100*5 + 21*10 + 2*10) / 365 = 730/365 = 2