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If Sigma(r=1)^(n) r^4=I(n), " then "Sigm...

If `Sigma_(r=1)^(n) r^4=I(n), " then "Sigma__(r=1)^(n) (2r -1)^4` is equal to

A

`I(2n)-I(n)`

B

`I(2n)-16 I(n)`

C

`I(2n)-8I(n)`

D

`I(2n)-4I(n)`

Text Solution

Verified by Experts

The correct Answer is:
B
`I(2n)=1^(4)+2^(4)+3^(4)+…+(2n-1)^(4)+(2n)^(4)`
`=[(1^(4)+3^(4)+5^(4)+..+(2n-1)^(4)]+2^(4)(1^(4)+2^(4)+3^(4)+4^(4)+…n^(4))`
`=sum_(r=1)^(n)(2r-1)^(4)+16xxI(n)`
`rArrsm_(r=1)^(n)(2r-1)^(4)=I(2n)-16I(n)`
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