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Q.

If $A_{1}, A_{2}, A_{3}, G_{1}, G_{2}, G_{3}$ and $H_{1}, H_{2}, H_{3}$ are three $A . M ., G . M .,$ and $H . M .$ between two positive numbers a and $b (a > b),$ then which of the following is true

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a

$2 G_{1} A_{1} = H_{1}^{2} (A_{1} + A_{3})$

b

$2 G_{1} G_{3} = H_{2} (A_{1} + A_{3})$

c

$A_{2} H_{2} = G_{2}^{2}$

d

$A_{2} G_{2} = H_{2}^{}$

answer is A, B.

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Detailed Solution

$A_{1} = \frac{3 a + b}{4}, A_{2} = \frac{a + b}{4}, A_{3} = \frac{a + 3 b}{4} G_{1} = {(a^{3} b)}^{1 / 4}, G_{2} = {(a b)}^{1 / 2}, G_{3} = {(a b^{3})}^{1 / 4} H_{1} = \frac{4 a b}{a + 3 b}, H_{2} = \frac{2 a b}{a + b}, H_{3} = \frac{4 a b}{3 a + b}$

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