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=(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} \\ (AH)/(A)=(ab)/(A) \\ H=(ab)/(A) \end{gathered}`

Substituting for A into the above expression,

`\begin{gathered} \text{recall A=}(a+b)/(2) \\ H=(ab)/(A)=(ab)/((a+b)/(2)) \\ H=(2*(ab)/(a+b))=(2ab)/(a+b) \\ H=(2ab)/(a+b) \end{gathered}`

Hence, A is the correct option.