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.

Examples

(4139) (Hockey Sticker Identity) Show that for any positive integer $n \ge k$, the following relationship holds: $$\binom{k}{k} +\binom{k+1}{k} + \binom{k+2}{k} + \cdots + \binom{n}{k} = \binom{n+1}{k+1} $$ |

(2690) Find the remainder when $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2018\times 2019$ is divided by $2020$. |