First slide

Theory of equations

Question

If $x = 1 + \frac{1}{3 + \frac{1}{2 + \frac{1}{3 + \frac{1}{2 \dots \infty}}}}$ , then value of x is

Moderate

A

$\sqrt{\frac{5}{2}}$
B

$\sqrt{\frac{3}{2}}$
C

$\sqrt{\frac{7}{3}}$
D

$\sqrt{\frac{5}{3}}$

Solution

$x = 1 + \frac{1}{3 + \frac{1}{2 + \frac{1}{3 + \frac{1}{2 \dots \infty}}}}$
$\begin{array}{l} = 1 + \frac{1}{3 + \frac{1}{1 + x}} \\ = 1 + \frac{1 + x}{4 + 3 x} \end{array}$
or $3 x^{2} = 5$
or $x = \sqrt{\frac{5}{3}}$ (as x is positirre)

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