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JEE Advanced 2007Paper 1

Exam: JEE Advanced
Year: 2007
Paper: Paper 1
Subject: Mathematics
Chapter: Sequences and Series

JEE Advanced 2007 · Paper 1 · Mathematics · Sequences and Series

Paragraph: Let V_r denotes the sum of the first r terms of an...

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?

Options

A $Q_1, Q_2, Q_3, \ldots$ are in AP with common difference 5
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 $Q_1=Q_2=Q_3=\ldots$

Correct answer

B. $Q_1, Q_2, Q_3, \ldots$ are in AP with common difference 6

Step-by-step 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

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