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The value of sum(r=1)^(n)(-1)^(r-1)((r )...

The value of `sum_(r=1)^(n)(-1)^(r-1)((r )/(r+1))*^(n)C_(r )` is (a) `1/(n+1)` (b) `1/n` (c) `1/(n-1)` (d) 0

A

`(1)/(n+1)`

B

`(1)/(n)`

C

`(1)/(n-1)`

D

`0`

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `(r )/(r+1)^(n)C_(r )=(1-(1)/(r+1))^(n)C_(r )`
`=^(n)C_(r )-(1)/(r+1)*(n!)/(r!(n-r)!)`
`=^(n)C_(r )-(1)/(n+1)'^(n+1)C_(r+1)`
`:.sum_(r=1)^(n)(-1)^(r-1)(r )/(r+1)*^(n)C_(r )`
`=^(n)C_(1)-^(n)C_(2)+C_(3)-.....-(1)/(n+1)['^(n+1)C_(2)-^(n+1)C_(3)+^(n+1)C_(4)-.....]`
`=^(n)C_(0)-(1)/(n+1)[-^(n+1)C_(0)+^(n+1)C_(1)]`
`=1-(1)/(n+1)[-1+(n+1)]`
`=(1)/(n+1)`
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