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If (1 - x)^(n) = C(0) + C(1)x + C(2)x^(2...

If `(1 - x)^(n) = C_(0) + C_(1)x + C_(2)x^(2) + ......... + C_(n)x^(n)`, then the value of `1.C_(1) + 2.C_(3) + 3.C_(3) + ......... + n.C_(n) = `

A

a positive value

B

a negative value

C

divisible by `2^(n-1)`

D

divisible by `2^(n)`

Text Solution

Verified by Experts

The correct Answer is:
B, C
For `n = 2m`, the given expression is
`C_(0)-(C_(0)+C_(1))+(C_(0)+C_(1)+C_(2))-(C_(0)+C_(1)+C_(2)+C_(3))+"....."(-1)^(n-1)(C_(0)+C_(1)+"....."+C_(n-1))`
`= C_(0)-(C_(0)+C_(1))+(C_(0)+C_(1)+C_(2))-(C_(0)+C_(1)+C_(2)+C_(3))+"....."-(C_(0)+C_(1)+".... "+C_(2m-1))`
`= - (C_(1)+C_(3)+C_(5)+"...."+C_(2m-1))`
`= - (C_(1)+C_(3)+C_(5)+"......."+C_(n-1)) = -2^(n-1)`
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