If one geometric G and two arithmetic means A1 and A2 are inserted between two distinct positive numbers, then 2A1−A2G2A2−A1G equal to

If one geometric $G$ and two arithmetic means $A_{1}$ and $A_{2}$ are inserted between two distinct positive numbers, then $(\frac{2 A_{1} - A_{2}}{G}) (\frac{2 A_{2} - A_{1}}{G})$ equal to

Let two distinct positive numbers be $a$ and $b .$

We have

$2 A_{1} - A_{2} = 2 (a + d) - (a + 2 d) = a$

and $2 A_{2} - A_{1} = 2 (b - d) - (b - 2 d) = b$

Thus, $(\frac{2 A_{1} - A_{2}}{G}) (\frac{2 A_{2} - A_{1}}{G}) = \frac{a b}{G^{2}} = 1$

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