Let A 1 , G 1 , H 1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n 2 , let A n - 1 and H n - 1 have arithmetic, geometric and harmonic means as A n , G n , H n respectively. Q. Whic

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

(3)

$Given A_{1} = \frac{a + b}{2}, G_{1} = \sqrt{a b}, H_{1} = \frac{2 a b}{a + b}$

$Also A_{n} = \frac{A_{n - 1} + H_{n - 1}}{2}, G_{n} = \sqrt{A_{n - 1} H_{n - 1}}$

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

$\Rightarrow G_{n}^{2} = A_{n} H_{n} \Rightarrow A_{n} H_{n} = A_{n - 1} H_{n - 1}$

$Similarly we can prove$

$A_{n} H_{n} = A_{n - 1} H_{n - 1} = A_{n - 2} H_{n - 2} = \dots = A_{1} H_{1}$

$\Rightarrow A_{n} H_{n} = a b$

$∴$ $G_{1}^{2} = G_{2}^{2} = G_{3}^{2} = \dots = a b$

$\Rightarrow G_{1} = G_{2} = G_{3} = \dots = \sqrt{a b}$

Solution

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

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

3 $G_{1} = G_{2} = G_{3} = \dots$

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

Want Us to Email you About Special Offers & Updates?

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 G1>G2>G3>⋯

2 G1<G2<G3<⋯

3 G1=G2=G3=⋯

4 G1<G3<G5<⋯ and G2>G4>G6>⋯

Ans. (3) Given A1=a+b2, G1=ab, H1=2aba+b Also An=An-1+Hn-12, Gn=An-1Hn-1 Hn=2An-1Hn-1An-1+Hn-1 ⇒Gn2=AnHn⇒AnHn=An-1Hn-1 Similarly we can prove AnHn=An-1Hn-1=An-2Hn-2=⋯=A1H1 ⇒AnHn=ab ∴ G12=G22=G32=⋯=ab ⇒G1=G2=G3=⋯=ab

Want Us to Email you About Special Offers & Updates?

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

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

3 $G_{1} = G_{2} = G_{3} = \dots$

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