prove for r, n ∈ N Σ_(k=r) ^n ((k),(r) ) = (((n+1)),((r+1)) ) (Hockey−stick identity)


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Question Number 185659 by mr W last updated on 25/Jan/23

$p r o v e f o r r, n \in N$ $\underset{k = r}{\sum^{n}} (\begin{matrix} k \\ r \end{matrix}) = (\begin{matrix} n + 1 \\ r + 1 \end{matrix})$ $(H o c k e y - s t i c k i d e n t i t y)$
Commented by mr W last updated on 25/Jan/23

	Answered by mr W last updated on 25/Jan/23

	$(\begin{matrix} n + 1 \\ r + 1 \end{matrix})$ $= (\begin{matrix} n \\ r \end{matrix}) + (\begin{matrix} n \\ r + 1 \end{matrix})$ $= (\begin{matrix} n \\ r \end{matrix}) + (\begin{matrix} n - 1 \\ r \end{matrix}) + (\begin{matrix} n - 1 \\ r + 1 \end{matrix})$ $= (\begin{matrix} n \\ r \end{matrix}) + (\begin{matrix} n - 1 \\ r \end{matrix}) + (\begin{matrix} n - 2 \\ r \end{matrix}) + (\begin{matrix} n - 2 \\ r + 1 \end{matrix})$ $. . . . . .$ $= (\begin{matrix} n \\ r \end{matrix}) + (\begin{matrix} n - 1 \\ r \end{matrix}) + (\begin{matrix} n - 2 \\ r \end{matrix}) + . . . + (\begin{matrix} r + 1 \\ r \end{matrix}) + (\begin{matrix} r + 1 \\ r + 1 \end{matrix})$ $= (\begin{matrix} n \\ r \end{matrix}) + (\begin{matrix} n - 1 \\ r \end{matrix}) + (\begin{matrix} n - 2 \\ r \end{matrix}) + . . . + (\begin{matrix} r + 1 \\ r \end{matrix}) + (\begin{matrix} r \\ r \end{matrix})$ $= \underset{k = r}{\sum^{n}} (\begin{matrix} k \\ r \end{matrix})$
	Commented by cortano1 last updated on 25/Jan/23

	$y e s . . . g r e a t$
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