Question

The product of n positive numbers is unity. Then their sum is

• A. a positive integer
• B. divisible by n
• C. equals to $n+\frac{1}{n}$
• D. never less than n

Answer 1

The correct option is D never less than n

$\mathrm{Let}{x}_1,x_2,......,x_n\mathrm{be}\mathrm{the}n\mathrm{positive}\mathrm{numbers}.\mathrm{Given}\mathrm{that}{x}_1{x}_2......x_n=1\mathrm{We}\mathrm{know}\mathrm{for}\mathrm{positive}\mathrm{numbers},A.M.\ge G.M.\Rightarrow \frac{x_1+x_2+......+x_n}{n}\ge {\left(x_1{x}_2......x_n\right)}^{\frac{1}{n}}\mathrm{or},\frac{x_1+x_2+......+x_n}{n}\ge 1\mathrm{or},x_1+x_2+......+x_n\ge n$