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Let V(r) denote the sum of the first r t...

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`

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)="....."`

Text Solution

Verified by Experts

The correct Answer is:
B
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`
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