Let denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For let and have arithmetic, geometric and harmonic means as respectively. Q. Which one of the following statements is correct? [2007]

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 $H_{n - 1}$ have arithmetic, geometric and harmonic means as $A_{n}, G_{n}, H_{n}$ respectively. [2007]

Q. Which one of the following statements is correct

Q. Which one of the following statements is correct?

Smart Achievers · Accepted Answer

(1)

$A_{n} = \frac{A_{n - 1} + H_{n - 1}}{2}$

$∴$ $A_{n} - A_{n - 1} = \frac{A_{n - 1} + H_{n - 1}}{2} - A_{n - 1}$

$= \frac{H_{n - 1} - A_{n - 1}}{2} < 0 (∵ A_{n - 1} > H_{n - 1})$

$\Rightarrow$ $A_{n} < A_{n - 1} or A_{n - 1} > A_{n}$

$∴$ $A_{1} > A_{2} > A_{3} >$ ...

Solution

1 $A_{1} > A_{2} > A_{3} >$ ...

2 $A_{1} < A_{2} < A_{3} <$ ...

3 $A_{1} > A_{3} > A_{5} >$ ... and $A_{2} < A_{4} < A_{6} <$ ...

4 $A_{1} < A_{3} < A_{5} <$ ... and $A_{2} > A_{4} > A_{6} >$ ...

Ans.
(1)

$A_{n} = \frac{A_{n - 1} + H_{n - 1}}{2}$

$∴$ $A_{n} - A_{n - 1} = \frac{A_{n - 1} + H_{n - 1}}{2} - A_{n - 1}$

$= \frac{H_{n - 1} - A_{n - 1}}{2} < 0 (∵ A_{n - 1} > H_{n - 1})$

$\Rightarrow$ $A_{n} < A_{n - 1} or A_{n - 1} > A_{n}$

$∴$ $A_{1} > A_{2} > A_{3} >$ ...

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Solution

Q. 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 have arithmetic, geometric and harmonic means as An,Gn,Hn respectively. [2007] Q. Which one of the following statements is correct

1 A1>A2>A3>...

2 A1<A2<A3<...

3 A1>A3>A5>... and A2<A4<A6<...

4 A1<A3<A5<... and A2>A4>A6>...