Sequences And Series Ques 13
Passage
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, 8M)
- Which one of the following statements is correct?
(a) $G_1>G_2>G_{\mathrm{a}}>\ldots$
(b) $G_1<G_2<G_3<\ldots$
(c) $G_1=G_2=G_3=\ldots$
(d) $G_1<G_3<G_5<\ldots$ and $G_2>G_4>G_6>\ldots$
Show Answer
Answer:
Correct Answer: 13.(c)
Solution: (c) Let $a$ and $b$ be two numbers. Then,
$ \begin{aligned} & A_1=\frac{a+b}{2} ; G_1=\sqrt{a b} ; H_1=\frac{2 a b}{a+b} \\ & A_n=\frac{A_{n-1}+H_{n-1}}{2}, \end{aligned} $
$ \begin{alignedat} 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}} \end{aligned} $
Clearly, $G_1=G_2=G_3=\ldots=\sqrt{a b}$.
BETA
