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AIEEE 2002

JEE Main 2002 Mathematics question on Sequences and Series

If a1,a2,…,ana_1, a_2, \ldots, a_n are positive real numbers whose product is a fixed number cc, then the minimum value of a1+a2+…+an−1+2ana_1 + a_2 + \ldots + a_{n-1} + 2a_n is
  1. A.
    n(2c)1/nn(2c)^{1/n}
  2. B.
    (n+1)c1/n(n+1)c^{1/n}
  3. C.
    2nc1/n2nc^{1/n}
  4. D.
    (n+1)(2c)1/n(n+1)(2c)^{1/n}
Show answer and solution

Correct option: A

Solution

By AM-GM inequality applied to the n terms a₁, a₂, ..., an-1, 2aₙ:

(a₁ + a₂ + ... + an-1 + 2aₙ)/n ≥ (a₁ · a₂ · ... · an-1 · 2aₙ)^(1/n)

The product inside is 2 · (a₁a₂...aₙ) = 2c.

So the sum ≥ n(2c)^(1/n), with equality when all n terms are equal, i.e., a₁ = a₂ = ... = an-1 = 2aₙ, which is achievable. The minimum value is n(2c)^(1/n).

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