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Binomial Sum

Source: 2008 SDMO Middle School Problem 2

August 24, 2016

Problem Statement

Find the sum of the first 5555 terms of the sequence \binom{0}{0}, \binom{1}{0}, \binom{1}{1}, \binom{2}{0}, \binom{2}{1}, \binom{2}{2}, \binom{3}{0}, \ldots. Note: For nonnegative integers nn and kk where 0≤k≤n0\leq k\leq n, (nk)=n!k!(n−k)!.\binom{n}{k}=\frac{n!}{k!\left(n-k\right)!}.
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