Binomials (1 Viewer)

[Hotdog1]

Hi, has anyone done these questions from the Fitzpatrick 3U book? Ex 29(a)?

Q50
by considering the coefficinets of
1+(1+x)+(1+x)^2+...+(1+x)^n,
prove that [nCr]+[(n-1)Cr]+[(n-2)Cr]+...+[rCr] = [(n+1)C(r+1)]
similarly consdering
x^n+[x^(n-1)](1+x)+[x^(n-2)](1+x)^2+...+(1+x)^n,
find the sum [nCr]+[(n-1)C(r-1)]+[(n-2)C(r-2)]+...+[(n-r-1)C1]

Q51
If P_r = [nCr]*x^r*(1-x)^(n-r),
Show that P_1 + 2P_2 +3P_3 + ... + nP_n =nx

Q52
By considering the coefficients of x^2r *(2r<n) in the expansion of (1+x)^n*(1-x)^n and (1-x^2)^n,
show that
[nC(r-1)][nC(r+1)] - [nC(r-2)][nC(r+2)] + ... + (-1)^r*[nC1][nC(2r-1)] - (-1)^r*[nC2r] = (1/2)[nCr][(nCr)-1]

Thanks alot!

 

[Giant Lobster]

Q50:

use the geometric sum formula, with n+1 terms, a common ratio of (1+x) and first term being 1. Then get x^r from that and compare it with x^r from that 1 + (1+x) + ... but write it backwards so u start with (1+x)^n.

Q 51,
U have to express P_1 + P_2 + ... in sigma notation, then as a binomial minus P_0 because it was unaccounted for. use the result r(nCr) = n((n-1)C(r-1)) so u can take the n out of the sigma notation. Then u fiddle around with the stuff inside the sigma, (Im just writing from memory cos i did this not long ago) and u shud end up with the answer.

cant do 52

 

[kimmeh]

ouch.. binomial bites me in the leg bad... i hate it.. theres like no set way to do it.. u have to use ur imagination

 

[Hotdog1]

Can you please go through that again? I don't think I understand. Thank you.

 

[jogloran]

Aargh, for Q50, I managed to get the LHS of the proof, but I keep getting something different for the general term using the GP. Help!

 

[Dash]

I would help.... but I always have trouble reading equations that are typed out

 

[freaking_out]

Originally posted by Dash
I would help.... but I always have trouble reading equations that are typed out

same here, thats why u hardly c me answering questions, coz i can't b bothered to translate the typed up stuff into proper maths.