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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 VrV_r denotes the sum of the first rr terms of an arithmetic progression (AP)(A P) whose first term is rr and the common difference is (2r−1)(2 r-1). Let Tr=Vr+1−Vr−2T_r=V_{r+1}-V_r-2 and Qr=Tr+1−TrQ_r=T_{r+1}-T_r for r=1,2,…r=1,2, \ldotsQuestion: TrT_r is always

Options

  1. Aan odd number
  2. Ban even number
  3. Ca prime number
  4. Da composite number

Correct answer

D. a composite number

Step-by-step solution

V_{r+1}-V_r=(r+1)^3-r^3-\frac{1}{2}\left[(r+1)^2-r^2\right]+\frac{1}{2}(1) =3r2+2r+1∴Tr=3r2+2r−1=(r+1)(3r−1)\begin{aligned}& =3 r²+2 r+1 \\ \therefore \quad T_r & =3 r²+2 r-1=(r+1)(3 r-1) \end{aligned} which is a composite number.
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