"Counting in their heads" - 1895 oil painting
08-09-2020, 12:19 PM (This post was last modified: 08-09-2020 12:37 PM by Albert Chan.)
Post: #5
 Albert Chan Senior Member Posts: 1,897 Joined: Jul 2018
RE: "Counting in their heads" - 1895 oil painting
(08-08-2020 07:42 PM)Albert Chan Wrote:  $$F(x) = \sum_{t=0}^{x-1} f(t) = 100\binom{x}{1} + 21\binom{x}{2} + 2\binom{x}{3}$$

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

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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