Exercise · ZW-CONTINUED-FRACTION-007 · ★★☆ Intermediate
Compute the convergent \( p_2/q_2 \) of the continued fraction \( [4; 2, 6, 7] \).
Choose one option.
Show answer
- D. 58/13
Explanation
With \( a_2 = 6 \) the recurrences give \( p_2 = 6 \cdot 9 + 4 = 58 \) and \( q_2 = 6 \cdot 2 + 1 = 13 \). The option 9/2 is the previous convergent, and 415/93 is the full value.
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-006 Compute the convergent p_1/q_1 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.