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Let a and b be positive real numbers. If...

Let a and b be positive real numbers. If a, `A_(1), A_(2)`, b are in arthimatic progression, a `G_(1), G_(2),` b are in geometric progression and a, `H_(1), H_(2)`,b are in harmonic progression, show that `(G_(1)G_(2))/(H_(1)H_(2))=(A_(1)+A_(2))/(H_(1)+H_(2))=((2a+b)(a+2b))/(9ab)`.

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Since, `a, A_(1), A_(2),b` are in AP
`rArr A_(1) + A_(2) = a + b`
`a, G_(1), G_(2), b` are in GP `rArr G_(1) G_(2) = ab`
and `a, H_(1), H_(2), b` are in HP
`rArr H_(1) = (3ab)/(2b + a), H_(2) = (3ab)/(b +2a)`
`:. (1)/(H_(1)) + (1)/(H_(2)) = (1)/(a) + (1)/(b)`
`rArr (H_(1) + H_(2))/(H_(1)H_(2)) = (A_(1) + A_(2))/(G_(1) G_(2)) = (1)/(a) + (1)/(b)` ...(i)
Now, `(G_(1) G_(2))/(H_(1)H_(2)) = (ab)/(((3ab)/(2b +a)) ((3ab)/(b + 2b)))`
`= ((2a + b) (a + 2b))/(9ab)`...(ii)
From Eqs. (i) and (ii), we get
`(G_(1) G_(2))/(H_(1) H_(2)) = (A_(1) + A_(2))/(H_(1) + H_(2)) = ((2a + b)(a + 2b))/(9ab)`
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