Question

The product of n positive numbers is unity, then their sum is :

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

Answer 1

Answer: (D)

never less than n

Answer 2

Since, product of $n$ positive number is unity.
$\Rightarrow \; x_1 x_2 x_3 ... x_n = 1$ .....(i)
Using A.M. $\ge$ GM
$\Rightarrow \frac{x_{1} + x_{2} + ..... +x_{n}}{n} \ge \left(x_{1} x_{2} ..... x_{n}\right)^{\frac{1}{n}}$
$\Rightarrow x_{1} + x_{n} + ..... + x_{n} \ge n\left(1\right)^{\frac{1}{n}}$ [From $eq^n$ (i)]