The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\ldots . . \infty}}}}$ is equal to :

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Sequences and Series

The value of

3 + \frac{1}{4 + \frac{1}{3 + \frac{1}{4 + \frac{1}{3 + \dots ..\infty}}}}

is equal to :

Answer: (C)

Let

x

be the value of the continued fraction. The block

3 + \frac{1}{4 + \dots}

repeats after two levels, so

x = 3 + \frac{1}{4 + \frac{1}{x}} = 3 + \frac{x}{4 x + 1}

Clearing denominators gives

(x - 3) (4 x + 1) = x

, that is

4 x^{2} - 12 x - 3 = 0

, so

x = \frac{12 \pm 144 + 48}{8} = \frac{12 \pm 8 3}{8} = \frac{3 \pm 2 3}{2}

The continued fraction exceeds

3

, so the root with the plus sign applies and the value is

\frac{3 + 2 3}{2} = 1.5 + 3

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Single correct option