If `sum_(r=0)^n{a_r(x-alpha+2)^r-b_r(alpha-x-1)^r}=0,` then prove that `b_n-(-1)^n a_n=0.`

Options

A. `b_(n) = 1+a_(n)`
B. `b_(n) = (-1)^(n)xxa_(n)`
C. `b_(n) = (-1)^(n-1) xxa_(n)`
D. `b_(n) + 1 = a_(n)`

Answer

B

Explanation

Let `alpha - x - 1 = t`
Then `underset(r=0)overset(n)suma_(r)(1-t)^(r) = underset(r=0)overset(n)sumb_(r)t^(r)`
`rArr b_(a)` = coefficient of `t^(n)` in `underset(r=0)overset(n)sum a_(r)(1-t)^(r)`
`=` coefficient of `t^(n)` in `a_(n)(1-t)^(n)`
`= (-1)^(n) xx a_(n)`