Question

The coefficient of `x^(n)` in the expansion of `(a+bx+cx^(2))/(e^(x))` is
A. `(-1)^(n)/(n!){cn^(2)-(b+c)(n+a}`
B. `(-1)^(n)/(n!){cn^(2)+(b+c)(n+a}`
C. `(-1)^(n)/(n!){cn^(2)+(b+c)(n-a}`
D. none of these

Answer 1

Answer:
We have
`(a+bx+cx^(2))/(e^(x))=ae^(-x)+bxe^(-x)+cx^(2)e^(-x)`
=coefficient of `x^(n) "in" (a+bx+cx^(2)+ex)`
=coefficient of `x^(n) in (ae^(-x)+bxe^(-x)+cx^(2)e^(-x)`
`=a("coeff of" x^(n)) in e^(-x))+b("coeff of" x^(n-1)) in e^(-x)+("coeff of" x^(n)-2) in e^(-x))`
`=a(-1)^(n)/(n!)-bn(-1)^(n)/(n!)+c(-1)^(n)n(n-1)/(n!)`
`=(-1)^(n)/(n!){a-bn+cn(n-1)}`