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If sum(r=1)^(r=n)(r^(4)+r^(2)+1)/(r^(4)+...

If `sum_(r=1)^(r=n)(r^(4)+r^(2)+1)/(r^(4)+r)=(675)/(26)`, then `n` equal to

A

`10`

B

`15`

C

`25`

D

`30`

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )` `T_(e)=(r^(4)+r^(2)+1)/(r^(4)+r)=((r^(2)+r+1)(r^(2)-r+1))/(r(r+1)(r^(2)-r+1))=(r^(2)+r+1)/(r(r+1))`
`T_(r)=1+(1)/(r )-(1)/(r+1)`
`T_(1)=1+(1)/(1)-(1)/(2)`
`T_(2)=1+(1)/(2)-(1)/(3)`
`T_(3)=1+(1)/(3)-(1)/(4)`
………………..
`T_(n)=1+(1)/(n)-(1)/(n+1)`
`:.S_(n)=n+1-(1)/(n+1)=(675)/(26)`
`:.26(n+1)^(2)-26=675(n+1)`
`implies26(n+1)^(2)-675(n+1)-26=0`
`implies26(n+1)[n+1-26]+[(n+1)-26]=0`
`implies(n-25)(26n-27)=0`
`:. n=25`
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