Binomial Expansion Q53 fitzpatrick (1 Viewer)

[jesshika]

hey people.. im having trouble proving this binomial question


If a1, a2, a3, and a4 are the coefficients of any 4 consecutive terms in the expansion (1 + x)^n, prove that;


a1/(a1 + a2) + a3/(a3 + a4) = 2a2/(a2 + a3)


i expanded (1 + x)^n and then let the a1 a2 a3 a4 equal the first 4 terms of the expansion ( i.e. a1=1, a2=nC1, etc ... ) but still couldn't get it

 

[withoutaface]

Did you try going n!/[r!(n-r)!], and let r be an aribitary term, ie use r, r+1, r+2 etc?

 

[jesshika]

yeah i did the whole n!/[r!(n-r)!] then i kinda ended up with

2(n-1)(n+1)
---------------
n(n-2)

and when i did it again

i got

4
-------
n+1

so i aint entiresly sure


what dyu mean let r be an aribitary term ??? >.<"

 

[withoutaface]

As in a non-constant, because that way you end up with a more general proof than a specific case.

 

[jesshika]

but why would yu want a more general proof as opposed to a specific one ?
hahaha sorr i if im annoying ,,

 

[withoutaface]

Because the question asks to prove that if they are ANY four, so if you just picked a specific few then it might work for them, but not others.