Let A1, G1, H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let An−1 and Hn−1 has arithmetic, geometric and harmonic means as An, Gn, Hn respectively.
Which of the following statements is correct?

Question

a)
A1 > A2 > A3 > …
b)
A1 < A2 < A3 < …
c)
A1 > A3 > A5 > … and A2 < A4 < A6 < …
d)
A1 < A3 < A5 < … and A2 > A4 > A6 > …

Correct answer is option 'A'. Can you explain this answer?

Answer

We are given that the arithmetic mean of two positive numbers is $A_1$, the geometric mean is $G_1$, and the harmonic mean is $H_1$. We want to find the arithmetic mean, geometric mean, and harmonic mean of the numbers $A_1, G_1, H_1$.

Let $a$ and $b$ be the two positive numbers. We have the following equations:
\begin{align*}
\frac{a+b}{2}&=A_1 \
\sqrt{ab}&=G_1 \
\frac{2}{\frac{1}{a}+\frac{1}{b}}&=H_1
\end{align*}

We want to find the arithmetic mean, geometric mean, and harmonic mean of $A_1, G_1, H_1$. Let $A$, $G$, and $H$ be the arithmetic mean, geometric mean, and harmonic mean of $A_1, G_1, H_1$. We have the following equations:
\begin{align*}
\frac{A_1+G_1}{2}&=A \
\sqrt{A_1G_1}&=G \
\frac{2}{\frac{1}{A_1}+\frac{1}{G_1}}&=H
\end{align*}

We want to find the arithmetic mean, geometric mean, and harmonic mean of $A, G, H$. Let $A', G', H'$ be the arithmetic mean, geometric mean, and harmonic mean of $A, G, H$. We have the following equations:
\begin{align*}
\frac{A+G}{2}&=A' \
\sqrt{AG}&=G' \
\frac{2}{\frac{1}{A}+\frac{1}{G}}&=H'
\end{align*}

Finally, we want to find the arithmetic mean, geometric mean, and harmonic mean of $A', G', H'$. Let $A'', G'', H''$ be the arithmetic mean, geometric mean, and harmonic mean of $A', G', H'$. We have the following equations:
\begin{align*}
\frac{A'+G'}{2}&=A'' \
\sqrt{A'G'}&=G'' \
\frac{2}{\frac{1}{A'}+\frac{1}{G'}}&=H''
\end{align*}

We want to find $A''-A$, $G''-G$, and $H''-H$. Substituting the equations above, we have
$$A''-A=\frac{A'+G'}{2}-A=\frac{A'+G'-2A}{2}$$
$$G''-G=\sqrt{A'G'}-\sqrt{AG}=\sqrt{A'G'-AG}$$
$$H''-H=\frac{2}{\frac{1}{A'}+\frac{1}{G'}}-\frac{2}{\frac{1}{A}+\frac{1}{G}}=\frac{2}{\frac{1}{A'}+\frac{1}{G'}}-\frac{2}{\frac{1}{A}+\frac{1}{G}}$$
\[=\frac{2}{\frac{1

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