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Geometric Mean

If A 1, A 2 b...

Question

If $A_{1}, A_{2}$ be two AM'x and $G_{1}, G_{2}$ be two GM's between a and b, then find the value of $\frac{A_{1} + A_{2}}{G_{1} G_{2}}$

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Solution

$a, A_{1}, A_{2}$ b are in A.P.

$A_{1} - a = A_{2} - A_{1} = b - A_{2}$ ......... (1)

and $(a, G_{1}, G_{2},)$ b are in G.P.

$G_{1} = \sqrt{a G_{2}}, G_{2} = \sqrt{G_{1} b}$

$\Rightarrow G_{1} G - 2 = s q r t a b G_{1} G_{2}$

$\Rightarrow \sqrt{G_{1} G_{2}} = \sqrt{a b}$

$\Rightarrow G_{1} G_{2} = a b$ ........... (2)

From equation (1),

$2 A_{1} = A_{2} + a$

$2 A_{2} = A_{1} + b$

Adding these two,

$2 (A_{1} + A_{2}) = (A_{1} + A_{2}) + a + b$

$A_{1} + A_{2} = a + b$

$\Rightarrow \frac{A_{1} + A_{2}}{G_{1} G_{2}} = \frac{a + b}{a b}$

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