Question

The product of $n$ positive numbers is unity, then their sum will be

• A. a positive integer
• B. divisible by ${}^{\prime }{n}^{\prime }$
• C. equal to ${}^{\prime }{n^2}^{\prime }$
• D. never less than ${}^{\prime }{n}^{\prime }$

Answer 1

The correct option is D never less than ${}^{\prime }{n}^{\prime }$

Let $x_1,x_2,x_3,...,x_n$ be $n$ positive numbers
Now, $x_1{x}_2{x}_3...x_n=1$
Using $AM\ge GM$
$\Rightarrow \frac{x_1+x_2+x_3+...+x_n}{n}\ge (x_1{x}_2{x}_3...x_n{)}^{\frac{1}{n}}$
$\Rightarrow x_1+x_2+x_3+...+x_n\ge n$