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The value of sum(r=0)^(3n-1)(-1)^r .^(6...

The value of `sum_(r=0)^(3n-1)(-1)^r .^(6n)C_(2r+1)3^r` is

A

`2^(3n)`

B

`2^(2n-1)`

C

`2^(6n-1)`

D

`0`

Text Solution

Verified by Experts

The correct Answer is:
D
`(d)` `sum_(r=0)^(3n-1)(-1)^(r )'^(6n)C_(2r+1)3^(r )`
`="Imaginary part of" sum_(r=0)^(3n-1)(1)/(sqrt(3)i)*^(6n)C_(2r+1)(sqrt(3)i)^(2r+1)`
`=(1)/(sqrt(3)i)Im(1+sqrt(3))^(6n)`
`=(1)/(sqrt(3)i)Im(2^(6)(cos2pi+isin2pi))`
`=0`
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