What is the value of $\displaystyle \frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{3}}}} $ ?

$\dfrac{1}{3}$

$\dfrac{3}{7}$

$\dfrac{7}{11}$

$\dfrac{7}{10}$

$\dfrac{10}{3}$

Approach

Use your calculator, summing up what you can. You can also try manually doing it:

$$ \frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{3}}}} $$ $$ =\frac{1}{1+\frac{1}{1+\frac{1}{\frac{4}{3}}}} $$ $$ =\frac{1}{1+\frac{1}{1+\frac{3}{4}}} $$ $$ =\frac{1}{1+\frac{1}{\frac{7}{4}}} $$ $$ =\frac{1}{1+\frac{4}{7}} $$ $$ =\frac{1}{\frac{11}{7}} $$ $$ = \boxed{\frac{7}{11}} $$