Hockey-stick Identity Intermediate

Video tutorial

Lecture Notes

The Hockey-stick Identity is best to be remembered using Pascal triangle. The proof given in example $1$ (# 4139) is simply to express the explanation given in the video in an algebraic way. $${m \choose m} + {m +1 \choose m}+{m+2 \choose m}+\cdots+{n \choose m}={n+1 \choose m+1}$$

This is one of the most tested combinatorial identities in AIME level contests.


Note the video tutorial is the same as that in the lesson Basic Combinatorial Identities.



(Hockey Sticker Identity) Show that for any positive integer $n \ge k$, the following relationship holds: $$\binom{k}{k} +\binom{k+1}{k} + \binom{k+2}{k} + \cdots + \binom{n}{k} = \binom{n+1}{k+1} $$


Find the remainder when $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2018\times 2019$ is divided by $2020$.

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