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If the expansion in power of x of the fu...

If the expansion in power of x of the function
`(1)/(( 1 - ax)(1 - bx))` is `a_(0) + a_(1) x + a_(2) x^(2) + a_(3) x^(3) + …, ` then `a_(n)` is

A

`(b^(n)-a^(n))/(b-a)`

B

`(a^(n)-b^(n))/(b-a)`

C

`(a^(n+1)-b^(n+1))/(b-a)`

D

`(b^(n+1) - a^(n+1))/(b-a)`

Text Solution

Verified by Experts

The correct Answer is:
D
`(1)/((1-ax)(1-bx))=a_(0)+a_(1)x+a_(2)x^(2)+"....."+a_(n)x^(n)+"..."`
But `(1-ax)^(-1)(1-bx)^(-1) = (1+ax+a^(2)x^(2)+"....")xx(1-bx+b^(2)x^(2)+"...")`
`rArr` Coefficient of `x^(n)` is
`b^(n)+ab^(n-1)+a^(2)b^(n-2)+"...."+a^(n-1)b+a^(n)=(b^(n+1)-a^(n+1))/(b-a)`
`rArr a_(n) = (b^(n+1)-a^(n+1))/(b-a)`
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