If one G.M., G and two A.M's p and q be inserted between two given numbers, prove that
`G^(2)= (2p-q) (2q-p)`

Explanation

Let a and b the numbers
then ` . P = a + (b-a)/(3) = (2a + b)/(3)`
` q = a + (2(b-a))/(3) = (a + 2b)/(3)`
and ` G = sqrt(ab)`
Now , ` (2p -q) (2q -p) = ((4a + 2b)/(3) - (a + ab)/(3)) ((2a + 4b)/(3) - (2a + b)/(3))`
` = ((3a)/(3)) ((3b)/(3))`
` ab = (sqrt(ab))^(2) = G^(2)` .