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Question

If $$A_1 \, , \,A_2 \, ; \,G_1 \, , \,G_2$$ and  $$H_1,H_2$$  be two A.M.s, G.M.s and H.M.s between two quantities a and b then 
prove that $$A_1 \,  \,H_2 \, = \,A_2 \,  \,H_1 \, = \,G_1 G_2 \, = \, ab$$


Solution

$$a \ , \ A_1 \, , \, A_2 \, ,\,b $$ are in A.P.....(1)
$$a \ , \ H_1 \, , \, H_2 \, ,\,b $$ are in H.P.
$$\therefore \, \, \dfrac{1}{a} \, , \, \dfrac{1}{H_1} \, , \, \dfrac{1}{H_2} \, , \, \dfrac{1}{b}$$ are in A.P.
Multiply by ab.
$$\therefore \, \, b \, , \, \dfrac{ab}{H_1} \, , \, \dfrac{ab}{H_2} \, , \, a$$ are in A.P
Take in reverse order 
Or $$\therefore \, \, a \, , \, \dfrac{ab}{H_2} \, , \, \dfrac{ab}{H_a} \, , \, b$$ ......(2)
Compare (1) and (2) 
$$  A_1 \, = \, \dfrac{ab}{H_2} \, \,and \, \, A_2 \, = \, \dfrac{ab}{H_1}$$
$$\therefore \,\, A_1H_2 \, = \, A_2 H_1 \, = \, ab \, = \, G_1G_2$$

Mathematics

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