If `(x^(2) + x)/(1-x) = a_(1) x + a_(2) x^(2) + ...` to `infty , |x| lt 1,` then

Options

A. `a_(1) + a_(2) = 4`
B. `a_(1)- a_(2) = 3`
C. `a_(p) = a_(q)` for `p, q gt 1`
D. none of these

Answer

c

Explanation

We have,
`(x^(2) + x)/(1-x) = (x^(2) + x) (1- x)^(-1)`
`rArr (x^(2) + x)/(1-x) = (x^(2) - x) (1+x+ x^(2)+ ...)`
`rArr (x^(2) + x)/(1-x) = (x + x^(2) - x^(3) + ...) +(x^(2)+x^(3) + x^(4)+ ...)`
`rArr (x^(2) + x)/(1-x) =x + 2x^(2) + 2x^(3) +2x^(4) + ...`
`rArr x + 2x^(2) + 2x^(3) + 2x^(4) + ... = a_(1) x + a_(2) x^(2) + a_(3) x^(3) + ...`
`rArr a_(1) = 1, a_(r) = 2 ` for ` r= 2, 3 ,...`
`rArr a_(p) = a_(q) = 2` for ` p , q gt 1. `