Q41 / 66 · JEE Advanced 2007 (Paper 1) · Mathematics

Paragraph: Let $V_r$ denotes the sum of the first $r$ terms of an arithmetic progression $(A P)$ whose first term is $r$ and the common difference is $(2 r-1)$. Let $T_r=V_{r+1}-V_r-2$ and $Q_r=T_{r+1}-T_r$ for $r=1,2, \ldots$Question: Which one of the following is a correct statement?

  • A
    $Q_1, Q_2, Q_3, \ldots$ are in AP with common difference 5
  • Correct answer
    B
    $Q_1, Q_2, Q_3, \ldots$ are in AP with common difference 6
  • C
    $Q_1, Q_2, Q_3, \ldots$ are in AP with common difference 11
  • D
    $Q_1=Q_2=Q_3=\ldots$

Solution

Since, $T_r=3 r^2+2 r-1$ $$ \begin{array}{rlrl} & \therefore & T_{r+1} & =3(r+1)^2+2(r+1)-1 \\ & \therefore & Q_r & =T_{r+1}-T_r=3[2 r+1]+2[1] \\ \Rightarrow & Q_r & =6 r+5 \\ \Rightarrow & Q_{r+1} & =6(r+1)+5 \end{array} $$ Common difference $=Q_{r+1}-Q_r=6$

Hence, the correct answer is option: 'B'.

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