Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
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: The sum V1+V2+…+VnV_1+V_2+\ldots+V_n is

Options

  1. A112n(n+1)(3n2−n+1)\frac{1}{12} n(n+1)\left(3 n^2-n+1\right)
  2. B112n(n+1)(3n2+n+2)\frac{1}{12} n(n+1)\left(3 n^2+n+2\right)
  3. C12n(2n2−n+1)\frac{1}{2} n\left(2 n^2-n+1\right)
  4. D13(2n3−2n+3)\frac{1}{3}\left(2 n^3-2 n+3\right)

Correct answer

B. 112n(n+1)(3n2+n+2)\frac{1}{12} n(n+1)\left(3 n^2+n+2\right)

Step-by-step solution

Vr=r2[2r+(r−1)(2r−1)]=12(2r3−r2+r)∴ΣVr=12[2Σr3−Σr2+Σr]=12\lleft[2(n(n+1)2\rright)2−n(n+1)(2n+1)6+n(n+1)2]=\ffrac112n(n+1)(3n2+n+2). \begin{aligned}& V_r=\frac{r}{2}[2 r+(r-1)(2 r-1)]=\frac{1}{2}\left(2 r³-r²+r\right) \\ & \therefore \quad \Sigma V_r=\frac{1}{2}\left[2 \Sigma r³-\Sigma r²+\Sigma r\right] \\ & =\frac{1}{2} \lleft[2 \left(\frac{n(n+1)}{2}\rright)^2-\frac{n(n+1)(2 n+1)}{6}+\frac{n(n+1)}{2}\right]\\ & =\ffrac{1}{12}n(n+1) \left(3 n²+n+2\right)\text{. }\\ & \end{aligned}
Practice Sequences and Series on Quantrex Academy →

More from Sequences and Series

Let α = 3+4+8+9+13+14+… upto 40 terms. If ( β)^ (α)/(1020) is a root of the equation… JEE Main 2026The sum 1 + (1)/(2)(1^2 + 2^2) + (1)/(3)(1^2 + 2^2 + 3^2) + … upto 10 terms is equal to : JEE Main 2026The value of 1^3 - 2^3 + 3^3 - … + 15^3 is: JEE Main 2026The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a… JEE Main 2026For the functions f(θ) = α ^2θ + β ^2θ , and g(θ) = α ^2θ + β ^2θ , α > β > 0 , let _ 0 <… JEE Main 2026If the sum of the first 10 terms of the series (1)/(1 + 1^4 × 4) + (2)/(1 + 2^4 × 4) +… JEE Main 2026Let A_1, A_2, A_3, …, A_ 39 be 39 arithmetic means between the numbers 59 and 159 . Then… JEE Main 2026Let the sum of the first n terms of an A.P. be 3n^2 + 5n . Then the sum of squares of the… JEE Main 2026∑_ n=1 ^ 10 ( (528)/(n(n+1)(n+2)) ) is equal to: JEE Main 2026Let α=(1)/(4)+(1)/(8)+(1)/(16)+…∞ and β=(1)/(3)+(1)/(9)+(1)/(27)+…∞ . Then the value of… JEE Main 2026The first term of an A.P. of 30 non-negative terms is (10)/(3) . If the sum of this A.P.… JEE Main 2026Let a_1, a_2, a_3, … be an A.P. and g_1 = a_1, g_2, g_3, … be an increasing G.P. If a_1 =… JEE Main 2026The sum (1^3)/(1) + (1^3 + 2^3)/(1 + 3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) + ⋯ up to 8… JEE Main 2026Let α, α + 2, α ∈ ℤ , be the roots of the quadratic equation x(x+2) + (x+1)(x+3) +… JEE Main 2026Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common… JEE Main 2026If ∑_ k=1 ^ n a_k = 6n^3 , then ∑_ k=1 ^ 6 ( a_ k+1 - a_k 36 )^2 is equal to _______. JEE Main 2026 Full Sequences and Series list Sequences and Series JEE Maths PYQsJEE Mathematics PYQs All JEE Advanced PYQs