Exercise · ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-005 · ★☆☆ Basic
What is the value of \( p_1 \) for the continued fraction \( [4;3,2] \)?
Choose one option.
Show answer
- A. \( 13 \)
Explanation
With \( p_0=4 \) and \( p_{-1}=1 \), the recurrence gives \( p_1=a_1p_0+p_{-1}=3\cdot 4+1=13 \). The option \( 12 \) forgets to add \( p_{-1} \).
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-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] ?
- ZW-CONVERGENT-OF-A-CONTINUED-FRACTION-009 For [0;1,4,2] , what is the convergent p_3/q_3 ?
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