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Q.
The A. M. between two positive numbers $a$ and $b$ is twice the G. M. between them. The ratio of the numbers is
Sequences and Series
Solution:
Given $2 \sqrt{a b}=\frac{a+b}{2} \Rightarrow \sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=4$
$\Rightarrow t^{2}-4 t+5=0$, where $\sqrt{\frac{a}{b}}=t$
$\therefore t =2 \pm \sqrt{3}$
$ \Rightarrow \sqrt{\frac{ a }{ b }}=2 \pm \sqrt{3}$
$\therefore \frac{ a }{ b }=\frac{(2 \pm \sqrt{3})^{2}}{4-3}=\frac{(2 \pm \sqrt{3})^{2}}{(2)^{2}-(\sqrt{3})^{2}}$
$\therefore a : b =2+\sqrt{3}: 2-\sqrt{3}$
or $2-\sqrt{3}: 2+\sqrt{3}$