2022 AMC 10A Problem 1

Problem 1 of 25EasierAlgebra

What is the value of

3+\dfrac{1}{3+\dfrac{1}{3+\dfrac{1}{3}}}?

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Solution

We can simplify this expression as follows:

\begin{aligned} 3 +& \dfrac{1}{3 + \dfrac{1}{3 + \dfrac{1}{3}}} \\ &= 3 + \dfrac{1}{3 + \dfrac{1}{\dfrac{10}{3}}} \\ &= 3 + \dfrac{1}{3 + \dfrac{3}{10}} \\ &= 3 + \dfrac{1}{\dfrac{33}{10}} \\ &= 3 + \dfrac{10}{33} \\ &= \dfrac{109}{33}. \end{aligned}

Thus, D is the correct answer.

Concepts: continued fraction · fraction

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.