Exercise · ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-009 · ★★☆ Intermediate
For \( [0;1,4,2] \), what is the convergent \( p_3/q_3 \)?
Choose one option.
Show answer
- A. \( 9/11 \)
Explanation
With \( p_0=0, p_1=1 \) and \( q_0=1, q_1=1 \), we get \( p_2=4\cdot 1+0=4 \) and \( q_2=4\cdot 1+1=5 \); then \( p_3=2\cdot 4+1=9 \) and \( q_3=2\cdot 5+1=11 \), so the convergent is \( 9/11 \). The option \( 9/4 \) uses \( p_2 \) as denominator instead of \( q_3 \).
Review the concept: Convergent of a continued fraction
A convergent of a continued fraction \( [a_0; a_1, a_2, \ldots] \) is the rational number \( p_n/q_n \) obtained by truncating the expansion at the term \( a_n \).
\[\frac{p_n}{q_n}=[a_0;a_1,\ldots,a_n],\quad p_n=a_np_{n-1}+p_{n-2},\\q_n=a_nq_{n-1}+q_{n-2}\]
More practice problems: convergent of a continued fraction
- ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-001 What is the convergent p_0/q_0 of the continued fraction [3;7,16] ?
- ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-002 What is the value of q_1 for the continued fraction [2;5,4] ?
- ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-003 What is the convergent p_1/q_1 of [0;1,2,3] ?
- ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-004 What is the convergent p_2/q_2 of [1;2,2] ?
- ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-005 What is the value of p_1 for the continued fraction [4;3,2] ?
- ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-006 Compute the convergent p_2/q_2 of the continued fraction [1;3,2,4] .
- ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-007 If [0;2,3,4] has convergents p_n/q_n , what is q_3 ?
- ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-008 What is the value of p_3/q_3 for [2;1,1,2] ?
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