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 one of the following statements is correct?

Question

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 one of the following statements is correct? (A) A1 > A2 > A3 > ldots (B) A1 < A2 < A3 < ldots (C) A1 > A3 > A5 > ldots and A2 < A4 < A6 < ldots (D) A1 < A3 < A5 < ldots and A2 > A4 > A6 > ldots

Tardigrade Admin · Accepted Answer

Since A2 is the A.M. of A1 and H1 and A1 > H1, we get A1 > A2 > H1 (1) Also, A3 is the A.M. of A2 and H2 and A2 > H2. Therefore, A2 > A3 > H2 ⇒ A1 > A2 > A3 > ldots

$A_{1} > A_{2} > A_{3} > \dots$

$A_{1} < A_{2} < A_{3} < \dots$

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

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

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, $A_{2} > A_{3} > H_{2}$
$\Rightarrow A_{1} > A_{2} > A_{3} > \dots$

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​ has arithmetic, geometric and harmonic means as An​,Gn​,Hn​, respectively. Which one of the following statements is correct?

A1​>A2​>A3​>…

A1​<A2​<A3​<…

A1​>A3​>A5​>… and A2​<A4​<A6​<…

A1​<A3​<A5​<… and A2​>A4​>A6​>…

Since A2​ is the A.M. of A1​ and H1​ and A1​>H1​, we get A1​>A2​>H1​ (1) Also, A3​ is the A.M. of A2​ and H2​ and A2​>H2​. Therefore, A2​>A3​>H2​ ⇒A1​>A2​>A3​>…

$A_{1} > A_{2} > A_{3} > \dots$

$A_{1} < A_{2} < A_{3} < \dots$

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

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

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, $A_{2} > A_{3} > H_{2}$
$\Rightarrow A_{1} > A_{2} > A_{3} > \dots$