- #1

cientifiquito

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## Homework Statement

Let

*r*be a positive integer. For any number x, let

(x)

_{r}= x(x-1)(x-2)...(x-r+1)

Show that

(-1/2)

_{r}= (-1)

^{r}r!2

^{-2r}(2r take r)

## Homework Equations

by "2r take r" I mean what is usually denoted by (n / r) (written like a fraction but without the bar) and is calculated as: n!/(r!(n-r)!)

## The Attempt at a Solution

If I start from the definition of (x)

_{r}, plugging in -1/2, I get as far as:

(-1)

^{r}(-1/2)

^{r}(1)(1+2)(1+4)(1+6)...(1+2r-2)

i.e.,

(-1)

^{r}(-1/2)

^{r}(1)(1+2)(1+4)(1+6)...(2r - 1)

And if I start from what I'm supposed to be showing that (-1/2)

_{r}is equal to, I can get to

(-1)

^{r}(-1/2)

^{r}[(2r)!/(r!(2r-r)!)]

i.e.,

(-1)

^{r}(-1/2)

^{r}[(r+1)(r+2)...(2r)]

but obviously I'm not seeing the connection between the two