Question

Find the coefficient of x^10 in the expansion of 1+2x/1-2x^2

Answer 1

Answer:

        32

Step-by-step explanation:

Key fact:

• 1/(1 - u)  =  1 + u + u² + u³ + ...

So...

   (1 + 2x) / (1 - 2x²)

=  1/(1 - 2x²)  +  2x/(1 - 2x²)

=  1 + (2x²) + (2x²)² + ... + (2x²)⁵ + ... +

  + 2x (1 + (2x²) + (2x²)² + ... )

The terms from 1/(1 - 2x²) give even powers of x.

The terms from 2x/(1 - 2x²) give odd powers of x.

The only occurrence of x¹⁰ is the term (2x²)⁵ = 32x¹⁰.

So the coefficient of x¹⁰ is 32.