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Question

If a1a......an are positive real numbers whose product is a fixed number c, then the minimum value of a1+a2+.......+a(n-1)+2an is


A

n(2c)1/n

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B

(n+1)c1/n

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C

(2n)c1/n

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D

(n+1)2c1/n

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Solution

The correct option is A

n(2c)1/n


Calculate the minimum value of a1+a2+.......+a(n-1)+2an :

Given product, a1×a2×....×an=c

Multiplying both sides by 2 we get,

a1a2......(an-1)(2an)=2c.........(1)

We know AMGM

(a1+a2+a3+.+2an)na1a2.....(an-1)(2an)1/n(a1+a2+a3+.+2an)n(2c)1/n(fromequation(1))(a1+a2+a3+.+2an)n(2c)1/n

So the minimum value of a1+a2+.+an-1+2an=n(2c)1/n

Hence, the correct option is( A).


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