Sequences And Series Ques 11
- If $x_1, x_2, \ldots, x_n$ are any real numbers and $n$ is any positive integer, then
(1982, 1M)
(a) $n \sum_{i=1}^n x_i^2<\left(\sum_{i=1}^n x_i\right)^2$
(b) $n \sum_{i=1}^n x_i^2 \geq\left(\sum_{i=1}^n x_1\right)^2$
(c) $n \sum_{i=1}^n x_i^2 \geq n\left(\sum_{i=1}^n x_i\right)^2$
(d) None of these
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Answer:
Correct Answer: 11.(b)
Solution: (b) Since, $x_1, x_2, \ldots, x_n$ are positive real numbers.
$\therefore \quad $ Using $nth$ power mean inequality
$ \begin{aligned} & \frac{x_1^2+x_2^2+\ldots+x_n^2}{n} \geq\left(\frac{x_1+x_2+\ldots+x_n}{n}\right)^2 \\ & \Rightarrow \frac{n^2}{n}\left(\sum_{i=1}^n x_i^2\right) \geq\left(\sum_{i=1}^n x_i\right)^2 \Rightarrow n\left(\sum_{i=1}^n x_i^2\right) \geq\left(\sum_{i=1}^n x_i\right)^2 \\ \end{aligned} $
BETA
