Arithmetic & Geometric Mean

Question

one geometric mean $G$ and two arithmetic mean $p$ & $q$ are to be inserted between two numbers then prove that $G^2 = (2p-q)(2q-p).$

Answer 1

Let the two numbers be $a$ and $b$. Then

$$G^2=ab$$

Note that $p$ is 'the' arithmetic mean of $a$, $q$ and $q$ is `the' arithmetic mean of $p$ and $b$. So

$$2p=a+q \quad\text{and}\quad2q=p+b$$

The result follows.