a, g, h are the arithmetic mean,geometric mea | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given that $a, g, h$ are the arithmetic mean,geometric mean,and harmonic mean of two positive numbers $x$ and $y$ respectively.We have $a = \frac{

Vedclass · Accepted Answer

$g$ is the geometric mean between $a$ and $h$

Vedclass · Answer

(C) Given that $a, g, h$ are the arithmetic mean,geometric mean,and harmonic mean of two positive numbers $x$ and $y$ respectively.We have $a = \frac{x + y}{2}$,$g = \sqrt{xy}$,and $h = \frac{2xy}{x + y}$.Now,calculate $g^2$:$g^2 = (\sqrt{xy})^2 = xy$ ...$(i)$Next,calculate the product $ah$:$ah = \left(\frac{x + y}{2}\right) \cdot \left(\frac{2xy}{x + y}\right) = xy$ ...(ii)Comparing equations $(i)$ and (ii),we get $g^2 = ah$,which implies $g = \sqrt{ah}$.This confirms that $g$ is the geometric mean between $a$ and $h$.

$a, g, h$ are the arithmetic mean,geometric mean,and harmonic mean between two positive numbers $x$ and $y$ respectively. Identify the correct statement among the following:

Similar Questions

The sum of the series $1 + (1 + 2) + (1 + 2 + 3) + \dots$ up to $n$ terms will be:

If $a, b, c$ are in $H.P.$,then the value of $\left( \frac{1}{b} + \frac{1}{c} - \frac{1}{a} \right) \left( \frac{1}{c} + \frac{1}{a} - \frac{1}{b} \right)$ is

If $\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \dots + \infty = \frac{\pi^4}{90}$,then the value of $\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \dots + \infty$ is

Which number should be added to the numbers $13, 15, 19$ so that the resulting numbers are the consecutive terms of a $H.P.$?

If $x, y, z \in R^+$ such that $x + y + z = 4$,then the maximum possible value of $xyz^2$ is -

Vedclass Test Series

Exam Paper Generator

Online Exam Module