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Let V(r ) denotes the sum of the first r...

Let `V_(r )` denotes the sum of the first r terms of an arithmetic progression whose first term is r and the common difference is `(2r-1)`. Let `T_(r )=V_(r+1)-V_(r)-2" and " Q_(r )=T_(r+1)-T_(r )" for " r=1,2,"...."`
`T_(r )` is always

A

`(1)/(12)n(n+1)(3n^(2)-n+1)`

B

`(1)/(12)n(n+1)(3n^(2)+n+2)`

C

`(1)/(2)n(2n^(2)-n+1)`

D

`(1)/(3)(2n^(3)-2n+3)`

Text Solution

Verified by Experts

The correct Answer is:
B
`V_(r)=(r )/(2)[(2r+(r-1)(2r-1))]=(1)/(2)(2r^(3)-r^(2)+r)`
`:.sum_(r=1)^(n)V_(r)=(1)/(2)[2sum_(r=1)^(n)r^(3)-sum_(r=1)^(n)r^(3)+sum_(r=1)^(n)r]`
`=(1)/(2)[2((n(n+1))/(2))^(2)-(n(n+1)(2n+1))/(6)+(n(n+1))/(2)]`
`=(1)/(2)n(n+1)(3n^(2)+n+2)`
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