The value of 4+1 5+1 4+1 5+1 4+ is - Vidyadip

MCQ

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

✓
$2+\frac{2}{5} \sqrt{30}$
B
$2+\frac{4}{\sqrt{5}} \sqrt{30}$
C
$4+\frac{4}{\sqrt{5}} \sqrt{30}$
D
$5+\frac{2}{5} \sqrt{30}$

✓

Answer

Correct option: A.

$2+\frac{2}{5} \sqrt{30}$

a
$y =4+\frac{1}{\left(5+\frac{1}{ y }\right)}$

$y -4=\frac{ y }{(5 y +1)}$

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

$y =\frac{20+\sqrt{480}}{10}$

$y =\frac{20-\sqrt{480}}{10} \rightarrow$ rejected

$y =2+\sqrt{\frac{480}{100}}$

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