Question

Which of the following represenst the correct simplified value of the continued fraction

$x=1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}$

• A. 8/5
• B. 17/12
• C. 7/5
• D. 9/7

Answer 1

Answer:

B. 17/12

Simplify the continued fraction:

$x=1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}$

Work from the innermost fraction outward.

1. Innermost part:

$2+\frac{1}{2}=\frac{5}{2}$

2. Next level:

$2+\frac{1}{\frac{5}{2}}$
$=2+\frac{2}{5}$
$=\frac{12}{5}$

3. Finally:

$x=1+\frac{1}{\frac{12}{5}}$
$=1+\frac{5}{12}$
$=\frac{12}{12}+\frac{5}{12}$
$=\frac{17}{12}$

Therefore, the simplified value is

$\boxed{\frac{17}{12}}$