gen-define-continued-fractions-and-give-one-short-worked-example-7bf8

Define continued fractions and give one short worked example.

# Continued Fractions A **continued fraction** is an expression of the form: $$a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}}$$ This is often written compactly as $[a_0; a_1, a_2, a_3, \ldots]$ where $a_0$ is an integer and $a_1, a_2, a_3, \ldots$ are positive integers. ## Worked Example Convert $\frac{17}{5}$ to a continued fraction. **Step 1:** Divide 17 by 5: $$\frac{17}{5} = 3 + \frac{2}{5}$$ **Step 2:** Take the reciprocal of the fractional part: $$\frac{17}{5} = 3 + \cfrac{1}{\frac{5}{2}}$$ **Step 3:** Repeat with $\frac{5}{2}$: $$\frac{5}{2} = 2 + \frac{1}{2}$$ **Step 4:** Substitute back: $$\frac{17}{5} = 3 + \cfrac{1}{2 + \cfrac{1}{2}}$$ **Answer:** $\frac{17}{5} = [3; 2, 2]$ **Verification:** $3 + \cfrac{1}{2 + \cfrac{1}{2}} = 3 + \cfrac{1}{\frac{5}{2}} = 3 + \frac{2}{5} = \frac{17}{5}$ ✓