Question

I can enter the problem here so ill add a picture, sorry for the inconvenience

Answer 1

Given two numbers a and b

Their Arithmetic mean (A), Geometric mean (G) and Harmonic mean (H) are given below,

$\begin{gathered} A=\frac{a+b}{2} \\ G=\sqrt[]{ab} \\ G=\sqrt[]{AH} \end{gathered}$

To find the formula that correctly relates H, a and b,

Relating the last two equations, i.e the geometric and harmonic mean below,

$\begin{gathered} G=\sqrt[]{ab} \\ G=\sqrt[]{AH} \\ \text{relating both equations} \\ \sqrt[]{ab}=\sqrt[]{AH} \\ \text{Square both sides } \\ (\sqrt[]{AH})^2=(\sqrt[]{ab})^2 \\ AH=ab \\ \text{Make H the subject},\text{ by dividing both sides by A} \\ \frac{AH}{A}=\frac{ab}{A} \\ H=\frac{ab}{A} \end{gathered}$

Substituting for A into the above expression,

$\begin{gathered} \text{recall A=}\frac{a+b}{2} \\ H=\frac{ab}{A}=\frac{ab}{\frac{a+b}{2}} \\ H=(2\times\frac{ab}{a+b})=\frac{2ab}{a+b} \\ H=\frac{2ab}{a+b} \end{gathered}$

Hence, A is the correct option.