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

Answer: (A)

$n\, (2c)^{1/n}$

Answer 2

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}