NIMCET 2019 — Mathematics PYQ

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Question

NIMCET 2019 — Mathematics PYQ

NIMCET | Mathematics | 2019

If $\sum_{i = 1}^{n} x_{i} = 80$ and $\sum_{i = 1}^{n} x_{i}^{2} = 400$ , then a possible value of n among the following is

Choose the correct answer:

Explanation

CONCEPT:
The Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (RMS-AM-GM-HM),
is an inequality of the root-mean-square, arithmetic mean, geometric mean, and harmonic mean of a set of positive real numbers $x_{1}, x_{2}, \dots, x_{n}$ that says :
$\frac{x _{1}^{2} + \dots + x _{n}^{2}}{n} \geq \frac{x _{1} + \dots + x _{n}}{n} \geq n x_{1} \dots x_{n} \geq \frac{n}{\frac{1}{x _{1}} + \dots + \frac{1}{x _{n}}}$

CALCULATIONS:

Given $\sum_{i = 1}^{n} x_{i} = 80$ and $\sum_{i = 1}^{n} x_{i}^{2} = 400$

Since, root mean square ≥ arithmetic mean, then

$\frac{x _{1}^{2} + \dots + \dots + x _{n}^{2}}{n} \geq \frac{x _{1} + \dots + x _{n}}{n}$

$∴ \frac{400}{n} \geq \frac{80}{n}$

On solving this inequality, we get -

$\Rightarrow n \geq 16$