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Mean

Two AMs. A 1 ...

Question

Two AMs. $A_{1} a n d A_{2}$ , two GMs. $G_{1} a n d G_{2}$ and two HMs. $H_{1} a n d H_{2}$ are inserted between any two numbers. Then find the arithmetic mean between $H_{1} a n d H_{2}$ in terms of $A_{1}, A_{2}, G_{1}, G_{2} .$

\frac{A_{1} + A_{2}}{2 G_{1} G_{2}}

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\frac{A_{1} - A_{2}}{2 G_{1} G_{2}}

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\frac{A_{1} + A_{2}}{2 + G_{1} G_{2}}

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\frac{G_{1} + G_{2}}{2 A_{1} A_{2}}

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Solution

The correct option is A $\frac{A_{1} + A_{2}}{2 G_{1} G_{2}}$
$a, A_{1}, A_{2}, b a r e i n A . P . ∴ A_{1} - a = A_{2} - A_{1} = b - A_{2} A_{1} + A_{2} = a + b a, G_{1}, G_{2}, b a r e i n G . P . \Rightarrow \frac{G_{1}}{a} = \frac{G_{2}}{G_{1}} = \frac{b}{G_{2}} \Rightarrow G_{1} G_{2} = a b a, H_{1}, H_{2}, b a r e i n H . P . \Rightarrow \frac{1}{a}, \frac{1}{H_{1}}, \frac{1}{H_{2}}, \frac{1}{b} a r e i n A . P . \frac{1}{H_{1}} - \frac{1}{a} = \frac{1}{H_{2}} - \frac{1}{H_{1}} = \frac{1}{b} - \frac{1}{H_{2}}$

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Two AMs $A_{1}$ and $A_{2}$ , two GMs $G_{1}$ and $G_{2}$ and two HMs $H_{1}$ and $H_{2}$ are inserted between two numbers a and b, then

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\frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}}

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G_{1}, G_{2} : H_{1} H_{2} = A_{1} + A_{2} : H_{1} + H_{2}

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