If A1- A2 are the two arithmetic means between two numbers a and b and G1- G1 are two geometric means between same two numbers- then A1-A2G1G2-

If nbsp;A1, A2 are two arithmetic means between two numbers a and b and G1, G2 are geometric means between a and b. Figure i.e #160;A1=a+13b a=3a+b a3=2a+b3 ...

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If A1, A2 are...

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If A₁, A₂ are the two arithmetic means between two numbers a and b and G₁, G₁ are two geometric means between same two numbers, then $\frac{A_{1} + A_{2}}{G_{1} G_{2}} =$ _________.

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Solution

If A₁, A₂ are two arithmetic means between two numbers a and b and G₁, G₂ are geometric means between a and b.
Figure
$i . e A_{1} = a + \frac{1}{3} (b - a) = \frac{3 a + b - a}{3} = \frac{2 a + b}{3} and A_{2} = a + \frac{2}{3} (b - a) = \frac{3 a + 2 b - 2 a}{3} = \frac{a + 2 b}{3}$
Also G₁and G₂ are the geometric means between a and b.
$So, G = a {(\frac{b}{a})}^{\frac{1}{3}} = a^{\frac{2}{3}} b^{\frac{1}{3}} and G_{2} = a {(\frac{b}{a})}^{\frac{2}{3}} = a^{\frac{1}{3}} b^{\frac{2}{3}}$
$\begin{array}{rcl} \frac{A_{1} + A_{2}}{G_{1} G_{2}} & = & \frac{\frac{2 a + b}{3} + \frac{2 b + a}{3}}{a b} \\ = & \frac{2 a + b + 2 b + a}{3 a b} \\ = & \frac{3 (a + b)}{3 a b} \\ \frac{A_{1} + A_{2}}{G_{1} G_{2}} & = & \frac{a + b}{a b} \end{array}$

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If A1, A2 are the two arithmetic means between two numbers a and b and G1, G1 are two geometric means between same two numbers, then A1+A2G1G2=_________.

If A₁, A₂ are the two arithmetic means between two numbers a and b and G₁, G₁ are two geometric means between same two numbers, then $\frac{A_{1} + A_{2}}{G_{1} G_{2}} =$ _________.