"Counting in their heads"  1895 oil painting

08092020, 12:19 PM
(This post was last modified: 08092020 12:37 PM by Albert Chan.)
Post: #5




RE: "Counting in their heads"  1895 oil painting
(08082020 07:42 PM)Albert Chan Wrote: \(F(x) = \sum_{t=0}^{x1} f(t) = 100\binom{x}{1} + 21\binom{x}{2} + 2\binom{x}{3}\) Another way to get F(x), via Triangular numbers \(\sum_{x=1}^n \binom{x+0}{1} = \binom{n+1}{2} \) // triangular numbers \(\sum_{x=1}^n \binom{x+1}{2} = \binom{n+2}{3} \) // sum of triangular numbers 10² + 11² + 12² + 13² + 14² = 10² + (10²+21) + (10²+21+23) + (10²+21+23+25) + (10²+21+23+25+27) = 100*5 + 21*4 + 23*3 + 25*2 + 27*1 = 100*5 + 21*(4+3+2+1) + 2*(1+1+1+2+2+3) = 100*5 + 21*(1+2+3+4) + 2*(1+(1+2)+(1+2+3)) = \(100\binom{5}{1} + 21\binom{5}{2} + 2\binom{5}{3}\) This matched the quoted F(x) equation, thus gives the same answer 

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