Name: Finding N Arithmetic Means Between Two Numbers
Uploaded: 2019-07-19
Duration: 3 min 37 s
Description: When three numbers are in A.P., the middle term is called the arithmetic mean of the other two numbers. Watch this video for complete information.

Question

The arithmetic mean between two distinct positive numbers is twice the geometric mean between them. Find the ratio of greater to smaller.

Accepted Answer

Let the two distinct positive numbers be $a$ and $b$ where $a > b$ .

$A . M .$ between $a$ and $b$ $= \frac{a + b}{2}$

$G . M .$ between $a$ and $b$ $= \sqrt{a b}$

According to given condition, we have $\frac{a + b}{2} = 2 \sqrt{a b}$

$\Rightarrow \frac{a + b}{2 \sqrt{a b}} = \frac{2}{1}$

We know that, if $\frac{a}{b} = \frac{c}{d}$ then componendo and dividendo rule is $\frac{a + b}{a - b} = \frac{c + d}{c - d}$

Applying componendo and dividendo, we have $\frac{a + b + 2 \sqrt{a b}}{a + b - 2 \sqrt{a b}} = \frac{2 + 1}{2 - 1}$

$\Rightarrow \frac{(\sqrt{a} + \sqrt{b})^{2}}{(\sqrt{a} - \sqrt{b})^{2}} = \frac{3}{1}$

$\Rightarrow \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{3}}{1}$

Taking positive square root on both sides

$\frac{\sqrt{a} + \sqrt{b} + \sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b} - \sqrt{a} + \sqrt{b}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} [using componendo and dividendo]$

$\Rightarrow \frac{2 \sqrt{a}}{2 \sqrt{b}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}$

On squaring

$\Rightarrow \frac{a}{b} = \frac{(\sqrt{3} + 1)^{2}}{(\sqrt{3} - 1)^{2}} = \frac{3 + 1 + 2 \sqrt{3}}{3 + 1 - 2 \sqrt{3}} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}$

Thus, $a : b = (2 + \sqrt{3}) : (2 - \sqrt{3})$

The arithmetic mean between two distinct positive numbers is twice the geometric mean between them. Find the ratio of greater to smaller.

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