A) A positive integer
B) Equal to \[n+\frac{1}{n}\]
C) Divisible by \[n\]
D) Never less than
Correct Answer: D
Solution :
Given \[{{x}_{1}}.{{x}_{2}}.{{x}_{3}}..........{{x}_{n}}=1\] \[\because \] \[\text{A}\text{.M}\text{.}\,\ge \text{G}\text{.M}\text{.}\] \ \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+......+{{x}_{n}}}{n} \right)\,\ge \,{{({{x}_{1}}.{{x}_{2}}.{{x}_{3}}.......{{x}_{n}})}^{\frac{1}{n}}}\] \[={{(1)}^{\frac{1}{n}}}=1\] \ \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+........+{{x}_{n}}\ge n\] \ \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+.......+{{x}_{n}}\] can never be less than n.
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