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  4. If H1,H2 are two harmonic means between two positive numbers a and b(a≠ b),A and G are the arithmetic and geometric means between a and b, then (H2+H1/H2H1) is

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Q. If $ {{H}_{1}},{{H}_{2}} $ are two harmonic means between two positive numbers $a$ and $ b(a\ne b),A $ and $G$ are the arithmetic and geometric means between $a$ and $b$, then $ \frac{{{H}_{2}}+{{H}_{1}}}{{{H}_{2}}{{H}_{1}}} $ is

KEAMKEAM 2007Sequences and Series

Solution:

Since, $ {{H}_{1}} $ and $ {{H}_{2}} $ are two harmonic means between two positive numbers a and b, then
$ a,{{H}_{1}},{{H}_{2}},b $ are in HP.
$ \therefore $ $ {{H}_{1}}=\frac{3ab}{a+2b},{{H}_{2}}=\frac{3ab}{2a+b} $
Now, $ \frac{{{H}_{1}}+{{H}_{2}}}{{{H}_{1}}{{H}_{2}}}=\frac{1}{{{H}_{2}}}+\frac{1}{{{H}_{1}}} $
$=\frac{2a+b}{3ab}+\frac{a+2b}{3ab} $
$=\frac{3a+3b}{3ab}=\frac{a+b}{ab} $ .. (i)
Now, A is the arithmetic mean between a and b, then
$ 2A=a+b $ ...(ii)
and G is the geometric mean between a and b, then
$ ab={{G}^{2}} $ ...(iii)
From Eqs. (i), (ii) and (iii),
we get $ \frac{{{H}_{1}}+{{H}_{2}}}{{{H}_{1}}{{H}_{2}}}=\frac{2A}{{{G}^{2}}} $