Sequences And Series Ques 9
- The product of $n$ positive numbers is unity, then their sum is
(1991, 2M)
(a) a positive integer
(b) divisible by $n$
(c) equal to $n+\frac{1}{n}$
(d) never less than $n$
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Answer:
Correct Answer: 9.(d)
Solution: (d) Since, product of $n$ positive numbers is unity.
$ \Rightarrow \quad x_1+x_2-x_3 \ldots x_n=1 $ $\quad$ ……..(i)
Using $\mathrm{AM} \geq \mathrm{GM}, \frac{x_1+x_2+\ldots+x_n}{n} \geq\left(x_1 \cdot x_2 \ldots x_n\right)^{1 / n}$
$\Rightarrow \quad x_1+x_2+\ldots+x_n \geq n(1)^{1 / n} \quad$ [from Eq. (i)]
Hence, sum of $n$ positive numbers is never less than $n$.
BETA
