Question:
The product of n positive numbers is unity, then their sum is :
The product of n positive numbers is unity, then their sum is :
Updated On: Jan 30, 2025
- a positive integer
- divisible by n
- equal to $n + \frac{1}{n}$
- never less than n
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The Correct Option is D
Solution and Explanation
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)]
$\Rightarrow \; x_1 x_2 x_3 ... x_n = 1$ .....(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)]
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