Question

If $a_1+a_2,...,a_n$ are positive numbers such that $a_1\cdot a_2...a_n=1,$ then their sum is :

• A. A positive integer
• B. Divisible by $n$
• C. Never less than $n$
• D. None of these

Answer 1

Answer: (C)

Never less than $n$

$\frac{a_1+a_2+...a_n}{n}>(a_1\cdot a_2\cdot a_3\cdot ....a_n{)}^{\frac{1}{n}}$
$\Rightarrow a_1+a_2+....a_n>n\cdot 1(\because a_1\cdot a_2...a_n=1)$
$\Rightarrow$ The sum of $a_1^{\prime }s$ is never less than $n.$