Let a and b be two positive real numbers. SupposeA1, A2 are two arithmetic means; G1,G2 are two geometric means and H1,H2 are two harmonic means betweena and b, then  A1+A2H1+H2 is equal to

# Let $a$ and be two positive real numbers. Supposeare two arithmetic means; ${G}_{1},{G}_{2}$ are two geometric means and ${H}_{1},{H}_{2}$ are two harmonic means betweena and $b,$ then  $\frac{{A}_{1}+{A}_{2}}{{H}_{1}+{H}_{2}}$ is equal to

1. A

$\frac{2\left({a}^{2}+{b}^{2}\right)+5ab}{9ab}$

2. B

$\frac{{a}^{2}+{b}^{2}}{9ab}+5$

3. C

$\frac{{a}^{2}+{b}^{2}+5\left(a+b\right)}{9ab}$

4. D

$\frac{{a}^{2}+{b}^{2}+7\left(a+b\right)}{3\left(a+b\right)ab}$

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### Solution:

We have

${A}_{1}=a+\frac{1}{3}\left(b-a\right)=\frac{2a+b}{3}$

and ${A}_{2}=\frac{a+2b}{3}$

$\therefore {A}_{1}+{A}_{2}=a+b$

Also $\frac{1}{{H}_{1}}+\frac{1}{{H}_{2}}=\frac{1}{a}+\frac{1}{b}$

and $\frac{1}{{H}_{2}}=\frac{1}{3}\left(\frac{2a+b}{ab}\right)$

$⇒{H}_{1}+{H}_{2}=\frac{a+b}{ab}{H}_{1}{H}_{2}$

Therefore,

$\begin{array}{l}\frac{{A}_{1}+{A}_{2}}{{H}_{1}+{H}_{2}}=\frac{ab}{{H}_{1}{H}_{2}}=\frac{ab}{9}\frac{\left(a+2b\right)\left(2a+b\right)}{\left(ab{\right)}^{2}}\\ =\frac{2\left({a}^{2}+{b}^{2}\right)+5ab}{9ab}\end{array}$