Question

Find the coefficient of `x^7` in the expansion of `(1-x)^(-2)`?

Answer 1

The coefficient of `x^7` is `8`

Explanation:

The Binomial expansion of `(1+x)^n` is given by

`(1+x)^n=sum_(r=0)^(r=n)C_r^nx^r`,

where `C_r^n=(n!)/(r!(n-r)!)=(n(n-1)(n-2)...(n-r+1))/(1*2*3....*r)`

that is coefficients would be `1,n,(n(n-1))/(1*2),(n(n-1)(n-2))/(1*2*3),.....`

Hence `(1-x)^(-2)=sum_(r=0)^(r=n)C_r^(-2)(-x)^r`

i.e. `(1-x)^(-2)=1+((-2))/1(-x)+((-2)(-3))/(1*2)(-x)^2+((-2)(-3)(-4))/(1*2*3)(-x)^3+...`

and term containing `x^7` woud be

`((-2)(-3)(-4)(-5)(-6)(-7)(-8))/(1*2*3*4*5*6*7)(-x)^7`

= `8x^7`

And the coefficient of `x^7` is `8`