Question

Use binomial expansion to solve each exercise. (a) Find the coefficient of xy19 in (3x − 2y) 20.(b) Give a formula for the coefficient of x k in (x+1/ x) n .

Answer 1

9514 1404 393

Answer:

  (a) -29,884,416

  (b) C(n,(n -k)/2) . . . where C(n,k) = n!/(k!(n-k)!) and k ∈ ℕ

Step-by-step explanation:

(a) The coefficient of the k-th term of the expansion is ...

  C(20,k)(3x)^(20-k)(-2y)^k

For k=19, this is ...

  19(3x)(-2y)^19 = -29,884,416xy^19

The coefficient of the term is -29,884,416.

__

(b) The p-th term of the expansion of (x +1/x)^n is ...

  C(n,p)(x^(n-p)(1/x)^p = C(n,p)(x^(n-2p))

For some exponent k of x, the value of p will be ...

  k = n -2p

  p = (n -k)/2

Then the coefficient of x^k will be C(n, (n-k)/2), where C(n, x) = n!/(x!(n-x)!).

You will notice that there will only be a term x^k for k having the same parity as n.