The value of `sum_(r=0)^(50)(-1)^r(^(50)C_r)/(r+2)` is equal to `1/(50xx51)` b. `1/(52xx50)` c. `1/(52xx51)` d. none of these

Options

A. `((17)!-2^(16))/((17)!)`
B. `((18)!2^(17))/((18)!)`
C. `((16)!-2^(15))/((16)!)`
D. `((15)!-2^(14))/((15)!)`

Answer

A

Explanation

`(rxx2^(r))/((r+2)!) = ((r+2-2)2^(r))/((r+2)!)`
`= (2^(r))/((r+1)!)-(2^(r+1))/((r+2)!)`
`= - ((2^(r+1))/((r+2)!)-(2^(r))/((r+1)!))`
`= -(V(r)-V(r-1))`
`rArr underset(r=1)overset(15)sum(rxx2)/((r+2)!)=-(V(15)-V(0))`
`= - ((2^(16))/(17!)-(2)/(2!))`
`= 1-(2^(16))/((17)!)`