Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced 2010Paper 2
Exam
JEE Advanced
Year
2010
Paper
Paper 2
Subject
Mathematics
Chapter
Sequences and Series

JEE Advanced 2010 · Paper 2 · Mathematics · Sequences and Series

Let a_1, a_2, a_3, , a_ 11 be real numbers satisfying a_1=15,27-2...

Let a1,a2,a3,…,a11a_1, a_2, a_3, \ldots, a_{11} be real numbers satisfying a1=15,27−2a2>0a_1=15,27-2 a_2>0 and ak=2ak−1−ak−2a_k=2 a_{k-1}-a_{k-2} for k=3,4,…,11k=3,4, \ldots, 11. If a12+a22+…+a11211=90\frac{a_1^2+a_2^2+\ldots+a_{11}^2}{11}=90, then the value of a1+a2+…+a1111\frac{a_1+a_2+\ldots+a_{11}}{11} is equal to

Correct answer

0

Step-by-step solution

ak=2ak−1−ak−2⇒a1,a2,…,a11 are in AP ∴a12+a22+…+a11211=11a2+35×11d2+10ad11=90⇒225+35d2+150d=9035d2+150d+135=0⇒d=−3,−97\begin{aligned}& a_k=2 a_{k-1}-a_{k-2} \\ & \Rightarrow \quad a_1, a_2, \ldots, a_{11} \text{ are in AP }\\ & \therefore\frac{a_1^2+a_2^2+\ldots+a_{11}^2}{11}\\ &=\frac{11 a^2+35 \times 11 d^2+10 a d}{11}=90 \\ & \Rightarrow \quad 225+35 d²+150 d=90 \\ & 35 d²+150 d+135=0 \\ & \Rightarrow \quad d=-3,-\frac{9}{7} \end{aligned}Given, a2<272∴d=−3a_2 \lt \frac{27}{2} \therefore d=-3 and d≠−97d \neq-\frac{9}{7} \Rightarrow \quad a1+a2+…+a1111=112[\frac{a_1+a_2+\ldots+a_{11}}{11} =\frac{11}{2}[30-10 ×\times3]=0
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