The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\ldots \infty}}}}$ is :

(B) $4+\frac{4}{\sqrt{5}} \sqrt{30}$

(D) $2+\frac{4}{\sqrt{5}} \sqrt{30}$

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Sequences and Series

The value of

4 + \frac{1}{5 + \frac{1}{4 + \frac{1}{5 + \frac{1}{4 + \dots \infty}}}}

is :

Answer: (A)

Let

y

denote the value of the whole expression. The blocks

4

and

5

alternate, so the same expression

y

reappears two levels down:

y = 4 + \frac{1}{5 + \frac{1}{y}} = 4 + \frac{y}{5 y + 1} .

Clearing the denominator gives

y (5 y + 1) = 4 (5 y + 1) + y

, that is

5 y^{2} - 20 y - 4 = 0

, so

y = \frac{20 \pm 400 + 80}{10} = \frac{20 \pm 4 30}{10} .

The expression is larger than

4

, so the negative sign is rejected and the value is

2 + \frac{2}{5} 30

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Single correct option