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The sum of the series 1+(9)/(4)+(36)/(9)...

The sum of the series `1+(9)/(4)+(36)/(9)+(100)/(16)+…` infinite terms is

A

`446`

B

`746`

C

`546`

D

`846`

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` The given series can be written as `1^(3)+(1^(3)+2^(3))/(1+3)+(1^(3)+2^(3)+3^(3))/(1+3+5)`
`t_(n)=(1^(3)+2^(3)+3^(3)+.....n^(3))/(1+3+5+...+(2n-1))`
`=(n^(2)(n+1)^(2))/(4n^(2))=((n+1)^(2))/(4)`
`=(1)/(4)(n^(2)+2n+1)`
`:.S_(n)=(1)/(4)[sum_(k=1)^(n)k^(2)+2sum_(k=1)^(n)k+n]`
`:.S_(n)=(1)/(4)[(n(n+1)(2n+1))/(6)+n(n+1)+n]`
`:.S_(16)=(1)/(4)[(16.17.33)/(6)+16.17+16]`
`=(1)/(4)[88xx17+16xx17+16]=446`
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