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If a-1,a2, ,an are positive real number...

If `a-1,a_2, ,a_n` are positive real numbers whose product is a fixed number `c ,` then the minimum value of a1 + d2 +..... + an-1 + 2an is

A

`n(2c)^(1//n)`

B

`(n +1) c^(1//n)`

C

`2nc^(1//n)`

D

`(n + 1)(2c)^(1//n)`

Text Solution

Verified by Experts

The correct Answer is:
A
Given, `a_(1) a_(2) a_(3) ...a_(n) =c`
`rArr a_(1) a_(2) a_(3) ...(a_(n-1)) (2a_(n)) = 2c`...(i)
`:. (a_(1) + a_(2) + a_(3) + ...+ 2a_(n))/(n) ge (a_(1).a_(2).a_(3)...2a_(n))^(1//n)` [using AM `ge` GM]
`rArr a_(1) + a_(2) + a_(3) + ...+ 2a_(n) ge n(2c)^(1//n)` [from Eq. (i)]
`rArr` Minimum value of
`a_(1) + a_(2) + a_(3) + ...+ 2a_(n) = n(2c)^(1//n)`
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