Q50 / 160 · AP EAMCET 2024 (19 May Shift 2) · Mathematics
$4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \infty}}}=$
- A$(2+\sqrt{5}),(2-\sqrt{5})$
- Correct answerB$2+\sqrt{5}$
- C$2-\sqrt{5}$
- D$2+\sqrt{3}$
Solution
Let, $x=4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \ldots . \infty}}}$
$\begin{aligned}
& \Rightarrow x=4+\frac{1}{x} \Rightarrow x^2-4 x-1=0 \\
& \Rightarrow x=2 \pm \sqrt{5}
\end{aligned}$
$\Rightarrow x=2+\sqrt{5}$ and $x \neq 2-\sqrt{5}$ (It is negative).
Hence, the correct answer is option: 'B'.