Question

Let $A_{1}, G_{1}, H_{1}$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $n \geq 2$, let $A_{n-1}$ and $H_{n-1}$ has arithmetic, geometric and harmonic means as $A_{n}, G_{n}, H_{n}$, respectively.
Which one of the following statements is correct?

• A. $A_{1} > A_{2} > A_{3} > \ldots$
• B. $A_{1} < A_{2} < A_{3} < \ldots$
• C. $A_{1} > A_{3} > A_{5} > \ldots$ and $A_{2} < A_{4} < A_{6} < \ldots$
• D. $A_{1} < A_{3} < A_{5} < \ldots$ and $A_{2} > A_{4} > A_{6} > \ldots$

Answer 1

Answer: (A)

Since $A_{2}$ is the A.M. of $A_{1}$ and $H_{1}$ and $A_{1} > H_{1}$,
we get $A_{1} > A_{2} > H_{1}$ (1)
Also, $A_{3}$ is the A.M. of $A_{2}$ and $H_{2}$ and $A_{2} > H_{2}$.
Therefore, $A_{2} > A_{3} > H_{2}$
$\Rightarrow A_{1} > A_{2} > A_{3} > \ldots$