Inspired by an AIME problem

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Let $f(x)$ be the sum of the numbers from $1$ to $x$ , inclusive. Let $S$ be equal to $\displaystyle \sum_{x=1}^{2017} f(x)$ . What is the remainder when $S$ is divided by $1000$ ?

The answer is 969.

1 solution

Neil Patram
Sep 4, 2017

$f(x)$ is the same as $\binom{n+1}{2}$ . So, what the problem is asking for is $\binom{2}{2}\ + \binom{3}{2}\ + \binom{4}{2}\ + ... +\binom{2018}{2}$ . Using the Hockey Stick Theorem, we can see that this is equal to $\binom{2019}{3}$ which is equal to $1,369,657,969$ . The last three digits are $\boxed{969}$

Inspired by an AIME problem

The answer is 969.

1 solution

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