How Do You Solve a Binomial Expansion Problem Involving Coefficients?

[CathyLou]

Hi.

I'm completely stuck on the following question, and have no idea how to even start it.

Any help would be really appreciated.

The first four terms, in ascending powers of x, of the binomial expansion of (1 + kx)^n are

1 + Ax + Bx^2 + Bx^3 + ...,

where k is a positive constant and A,B and n are positive intgers.

(a) By considering the coefficients of x^2 and x^3, show that 3 = (n - 2)k.

Thank you.

Cathy

 

[HallsofIvy]

Do you know the "binomial theorem":
$$(a+ b)^n= \sum_{i=0}^n _nC_i a^i b^{n-i}$$
where $_nC_i$ is the "binomial coefficient" n!/i!(n-i)!.

In particular, the coefficient if xⁱ in (1+ kx)^n is $_nC_i k^i$
Here, you are GIVEN that the coefficient of x², which is k²n(n-1)/2, and the coefficient of x³, which is k³n(n-1)(n-2)/6 are equal. Set them equal and cancel everything you can.

 

[CathyLou]

Okay. Thanks very much for your help!

Cathy