Exercise · ZW-CONTINUED-FRACTION-006 · ★★☆ Intermediate
Compute the convergent \( p_1/q_1 \) of the continued fraction \( [4; 2, 6, 7] \).
Choose one option.
Show answer
- A. 9/2
Explanation
With \( a_0 = 4 \) and \( a_1 = 2 \) we get \( p_1 = 2 \cdot 4 + 1 = 9 \) and \( q_1 = 2 \), so the convergent is \( 9/2 \). The values 4/1 and 58/13 correspond to other indices of the expansion.
Review the concept: Continued fraction
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
- ZW-CONTINUED-FRACTION-001 What is the term a_0 of the continued fraction of 415/93 ?
- ZW-CONTINUED-FRACTION-002 What is the value of the rational [2; 3] ?
- ZW-CONTINUED-FRACTION-003 How many terms does the continued fraction of 415/93 have according to the Euclidean algorithm?
- ZW-CONTINUED-FRACTION-004 Which rational number does the continued fraction [1; 1] represent?
- ZW-CONTINUED-FRACTION-005 What is the term a_1 of the continued fraction of 415/93 ?
- ZW-CONTINUED-FRACTION-007 Compute the convergent p_2/q_2 of the continued fraction [4; 2, 6, 7] .
- ZW-CONTINUED-FRACTION-008 What is the value of [0; 1, 2] ?
- ZW-CONTINUED-FRACTION-009 Write the continued fraction of 27/11 using the Euclidean algorithm.