Let `V_r` denote the sum of the first r terms of an arithmetic progression (AP) 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` Which one of the following is a correct statement? (A) `Q_1, Q_2, Q_3…………..,` are in A.P. with common difference 5 (B) `Q_1, Q_2, Q_3…………..,` are in A.P. with common difference 6 (C) `Q_1, Q_2, Q_3…………..,` are in A.P. with common difference 11 (D) `Q_1=Q_2=Q_3`

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`