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,"...."`
The sum V1​+V2​+...+Vn​ is

Options

A. `Q_(1),Q_(2),Q_(3),"....."` are in AP with common difference 5
B. `Q_(1),Q_(2),Q_(3),"....."` are in AP with common differemce 6
C. `Q_(1),Q_(2),Q_(3),"....."` are in AP with common difference 11
D. `Q_(1)=Q_(2)=Q_(3)="....."`

Answer

B

Explanation

Since, `T_(r )=3r^(2)+2r-1`
`:.T_(r+1)=3(r+1)^(2)+2(r+1)-1`
`:.Q_(r )=T_(r+1)-T_(r)=3[2r+1]+2[1]`
`implies Q_(r )=6r+5`
`implies Q_(r+1)=6(r+1)+5`
Common difference `=Q_(r+1)-Q_(r)=6`