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JEE Advanced 2007Paper 2
Exam
JEE Advanced
Year
2007
Paper
Paper 2
Subject
Mathematics
Chapter
Sequences and Series

JEE Advanced 2007 · Paper 2 · Mathematics · Sequences and Series

Paragraph: Let A_1, G_1, H_1 denote the arithmetic, geometric and...

Paragraph: Let A1,G1,H1A_1, G_1, H_1 denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For n≥2n \geq 2, let An−1A_{n-1} and Hn−1H_{n-1} has arithmetic, geometric and harmonic means as An,Gn,HnA_n, G_n, H_n, respectively.Question: Which of the following statements is correct?

Options

  1. AA1>A2>…A_1>A_2>\ldots
  2. BA1<A2<A3<…A_1 < A_2 < A_3 < \ldots
  3. CA1>A3>A5>…A_1>A_3>A_5>\ldots and A2<A4<A6<…A_2 < A_4 < A_6 < \ldots
  4. DA1<A3<A5<…A_1 < A_3 < A_5 < \ldots and A2>A4>A6>…A_2>A_4>A_6>\ldots

Correct answer

A. A1>A2>…A_1>A_2>\ldots

Step-by-step solution

Let f(x)=kex−xf(x)=k e^x-xf′(x)=kex−1=0x=−ln⁡k∴f′′(x)=kex Hence, [f′′(x)]x=−ln⁡k=1>0f(−ln⁡k)=1+ln⁡k\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 \gt 0 \\ & & f(-\ln k) & =1+\ln k\end{aligned}ForonerootofgivenequationFor one root of given equation\begin{aligned}1+\ln k & =0 \\ Hence, k & =\frac{1}{e} . \end{aligned}A_2$ is A M of $A_1 and H_1 and A_1 \gt H_1$ \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 \gt \mathrm{H}_2\begin{array}{ll} \Rightarrow& A_2>A_3>H_2 \\ \therefore & A_1>A_2>A_3>\ldots \end{array}$$$
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