Sum of a binomial, with differentiation? (1 Viewer)

[Michaelmoo]

How would you go about putting this question???

http://link.imgshare.us/b1b3xp

(sorry about the large image)

 

[Trebla]

x(1 + x)ⁿ = Σ ⁿC_r x^{r + 1}
Differentiate both sides with respect to x
(1 + x)ⁿ + nx(1 + x)^{n - 1} = Σ(r + 1) ⁿC_r x^r
Let x = -1
(1 + -1)ⁿ - n(1 + -1)^{n - 1} = Σ(r + 1) ⁿC_r (-1)^r
=> Σ(-1)^r (r + 1) ⁿC_r = 0

 

[Michaelmoo]

x(1 + x)ⁿ = Σ ⁿC_r x^{r + 1}
Differentiate both sides with respect to x
(1 + x)ⁿ + nx(1 + x)^{n - 1} = Σ(r + 1) ⁿC_r x^r
Let x = -1
(1 + -1)ⁿ - n(1 + -1)^{n - 1} = Σ(r + 1) ⁿC_r (-1)^r
=> Σ(-1)^r (r + 1) ⁿC_r = 0

Thanks. I figured It'd be 0. But it seemed weired for a 4 mark question to come out so nicely =\