Exercise · ZW-CONTINUED-FRACTION-010 · ★★★ Advanced
Let \( x = [1; 2, 2, 2] \). Compute \( x \) as a rational number and check that the convergent \( p_2/q_2 \) approximates it from below.
Choose one option.
A \( 17/12 \), with \( p_2/q_2 = 7/5 \) less than \( x \) B \( 12/17 \), with \( p_2/q_2 = 4/5 \) greater than \( x \) C \( 17/12 \), with \( p_2/q_2 = 5/4 \) greater than \( x \) D \( 17/12 \), with \( p_2/q_2 = 7/5 \) greater than \( x \) Show answer A. \( 17/12 \), with \( p_2/q_2 = 7/5 \) less than \( x \)← Back to the question
Explanation Working from the end: \( 2 + 1/2 = 5/2 \), then \( 2 + 2/5 = 12/5 \), and \( x = 1 + 5/12 = 17/12 \). The convergents are \( 1/1, 3/2, 7/5, 17/12 \), so \( p_2/q_2 = 7/5 \) lies below \( x \) because its index is even.
A continued fraction is an expression of the form a₀ + 1/(a₁ + 1/(a₂ + ...)), where a₀ is an integer and the remaining terms aᵢ are positive integers.
\[x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}}\]
More practice problems: continued fraction