If one geometric mean is $G$ and two arithmetic means are $p$ and $q$ , prove that : $G^{2} = (2 p - q) (2 q - p)$

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Solution

Let the two numbers are $a$ and $b$ ,
Geometric mean, $G = \sqrt{a b}$

To insert two arithmetic means between $a$ and $b$ ,
Common difference, $d = \frac{b - a}{3}$

$n$ th arithmetic mean $A_{n} = a + n d$

$p = a + d = a + \frac{b - a}{3}$

$q = a + 2 d = a + 2 \times \frac{b - a}{3}$

Now,
$(2 p - q) (2 q - p) = (a) (b)$

$(2 p - q) (2 q - p) = {(\sqrt{a b})}^{2} = G^{2}$

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