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The sum to 50 terms of the series 3/1...

The sum to 50 terms of the series
`3/1^2+5/(1^2+2^2)+7/(1^+2^2+3^2)+….+… is `

A

`100/17`

B

`150/17`

C

`200/51`

D

`50/17`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `T_(r)` be the rth term of the given series. Then,
`T_(r)=(2r+1)/(1^(2)+2^(2)+…+r^(2))=(6(2r+1))/((r )(r+1)(2r+1))`
`=6(1/r-1/(r+1))`
So, sum is given by
`sum_(r=1)^(50)T_(r)=6sum_(r=1)^(50)(1/r-1/(r+1))`
`=6[(1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/50-1/51)]`
`=6[1-1/51]=100/17`
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