Q40 / 66 · JEE Advanced 2007 (Paper 2) · Mathematics
Paragraph:
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.Question:
Which of the following statements is correct?
- Correct answerA$A_1>A_2>\ldots$
- 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 $A_2>A_4>A_6>\ldots$
Solution
Let $f(x)=k e^x-x$
$\begin{aligned} & & f^{\prime}(x) & =k e^x-1=0 \\ & & x & =-\ln k \\ & \therefore & f^{\prime \prime}(x) & =k e^x \\ & \text { Hence, } & {\left[f^{\prime \prime}(x)\right]_{x=-\ln k} } & =1>0 \\ & & f(-\ln k) & =1+\ln k\end{aligned}$
For one root of given equation
$$
\begin{aligned}
1+\ln k & =0 \\
Hence,
k & =\frac{1}{e} .
\end{aligned}
$$
$A_2$ is $A M$ of $A_1$ and $H_1$ and $A_1>H_1$
$$
\Rightarrow \quad A_1>A_2>H_1
$$
$A_3$ is $\mathrm{AM}$ of $\mathrm{A}_2$ and $\mathrm{H}_2$ and $\mathrm{A}_2>\mathrm{H}_2$
$$
\begin{array}{ll}
\Rightarrow & A_2>A_3>H_2 \\
\therefore & A_1>A_2>A_3>\ldots
\end{array}
$$
Hence, the correct answer is option: 'A'.