Question

Find the AM, GM and HM between $\frac{1}{2}$ and $\frac{1}{8}$

Answer 1

We know that the arithmetic mean between the two numbers $a$ and $b$ is $AM=\frac{a+b}{2}$, the geometric mean is $GM=\sqrt{ab}$ and the harmonic mean is $HM=\frac{2ab}{a+b}$.

Here, it is given that $a=\frac{1}{2}$ and $b=\frac{1}{8}$, therefore,

$AM=\frac{a+b}{2}=\frac{\frac{1}{2}+\frac{1}{8}}{2}=\frac{\frac{4+1}{8}}{2}=\frac{\frac{5}{8}}{2}=\frac{5}{8}\times \frac{1}{2}=\frac{5}{16}$

$GM=\sqrt{ab}=\sqrt{\frac{1}{2}\times \frac{1}{8}}=\sqrt{\frac{1}{16}}=\frac{1}{4}$

$HM=\frac{2ab}{a+b}=\frac{2\times \frac{1}{2}\times \frac{1}{8}}{\frac{1}{2}+\frac{1}{8}}=\frac{\frac{1}{8}}{\frac{4+1}{8}}=\frac{\frac{1}{8}}{\frac{5}{8}}=\frac{1}{5}$

Hence, $AM=\frac{5}{16}$, $GM=\frac{1}{4}$ and $HM=\frac{1}{5}$