Question

If the product of $n$ positive numbers is $1$, then their sum is

• A. A positive integer
• B. Divisible by $n$
• C. Equal to $n+\frac{1}{n}$
• D. Greater than or equal to $n$

Answer 1

The correct option is D Greater than or equal to $n$

Since, $A.M.\ge G.M.$

let n positive numbers be $x_1,x_2,...,x_n$

$A.M.=\frac{x_1+x_2+...+x_n}{n}$

$G.M.=x_1\ast x_2\ast ...\ast x_n$

Therefore, $\frac{x_1+x_2+...+x_n}{n}\ge \sqrt[n]{n{x}_1\ast x_2\ast ...\ast x_n}$

$⟹x_1+x_2+...+x_n\ge n\ast \sqrt[n]{n{x}_1\ast x_2\ast ...\ast x_n}$

$⟹x_1+x_2+...+x_n\ge n$