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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Sequences and Series

NDA 2019 · Mathematics · Sequences and Series

What is the value of 2+(1)/(2+(1)/(2+(1)/(2+ ∞)))

What is the value of 2+12+12+12+…∞2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\ldots \infty}}}

Options

  1. A2−1\sqrt{2}-1
  2. B2+1\sqrt{2}+1
  3. C3
  4. D4

Correct answer

B. 2+1\sqrt{2}+1

Step-by-step solution

Let y=2+12+12+…y=2+\frac{1}{2+\frac{1}{2+\ldots}}∴y=2+1y⇒y2=2y+1⇒y2−2y−1=0⇒y=2±(−2)2−4(−1)2⇒y=2±222=1±2∵y>2∴y=1±2\begin{aligned} & \therefore y=2+\frac{1}{y} \\ & \Rightarrow y^2=2 y+1 \\ & \Rightarrow y^2-2 y-1=0 \\ & \Rightarrow y=\frac{2 \pm \sqrt{(-2)^2-4(-1)}}{2} \\ & \Rightarrow y=\frac{2 \pm 2 \sqrt{2}}{2}=1 \pm \sqrt{2} \\ & \because \quad y > 2 \\ & \therefore \quad y=1 \pm \sqrt{2}\end{aligned}$$
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