Question

If $a_1,a_2,...,a_n$ are positive real numbers whose product is a fixed number c, then the minimum value of $ a_1 + a_2 +...+ a_{n-1}+2a_n$ is

• A. $ n\, (2c)^{1/n} $
• B. $ (n\, +1) c^{1/n} $
• C. $ 2cn^{1/n}$
• D. $ (n\, +1) (2c)^{1/n} $

Answer 1

The Correct Option is A

Given, $ a_1, a_2,...,a_n = c $
$\Rightarrow \, \, \, a_1\, a_2\, a_3...(a_{n-1})(2a_n) = 2c\hspace40mm ...(i)$
$\therefore \, \, \, \, \frac{a_1\, + \, a_2\, +\, a_3\, +...+2a_n}{n} \ge (a_1.a_2.a_3...2a_n)^{1/n}$
$ \hspace70mm [using\, AM \ge GM] $
$ \Rightarrow \, \, \, a_1+ a_2+ a_3+...+2a_n \ge \, n(2c)^{1/n} \hspace5mm [from\, E (i)] $
$ \Rightarrow \, \, \, $ Minimum value of
$ \hspace30mm a_1+ a_2+ a_3+...+2a_n = \, n(2c)^{1/n}$