Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced 2011Paper 1
Exam
JEE Advanced
Year
2011
Paper
Paper 1
Subject
Mathematics
Chapter
Sequences and Series

JEE Advanced 2011 · Paper 1 · Mathematics · Sequences and Series

The minimum value of the sum of real numbers a^ -5 , a^ -4 , 3 a^...

The minimum value of the sum of real numbers a−5,a−4,3a−3,1,a8a^{-5}, a^{-4}, 3 a^{-3}, 1, a^8 and a10a^{10} with a>0a>0 is

Correct answer

8

Step-by-step solution

Using AM≥GMA M \geq G M,a−5+a−4+a−3+a−3+a−3+1+a8+a108≥(a−5\cdota−4\cdota−3\cdota−3\cdota−3⋅1\cdota8\cdota10)18⇒a−5+a−4+3a−3+1+a8+a10≥8⋅1\begin{aligned}& \frac{a^{-5}+a^{-4}+a^{-3}+a^{-3}+a^{-3}+1+a^8+a^{10}}{8}\\ & \geq\left(a^{-5} \cdota^{-4} \cdota^{-3} \cdota^{-3} \cdota^{-3} \cdot1 \cdota^8 \cdota^{10}\right)^{\frac{1}{8}} \\ & \Rightarrow a^{-5}+a^{-4}+3 a^{-3}+1+a⁸+a^{10} \geq 8 \cdot1 \end{aligned}Hence, minimum value is 8 .
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