Exercise · ZW-RECURRENCE-RELATION-009 · ★★☆ Intermediate
With \(n\) lines in general position, the number of regions of the plane satisfies \(R_n=R_{n-1}+n\) and \(R_0=1\). What is \(R_6\)?
Choose one option.
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- C. 22
Explanation
\(R_6=1+1+2+3+4+5+6=22\), which matches \(1+n(n+1)/2\) for \(n=6\).
Review the concept: Recurrence relation
A recurrence relation is an equation that expresses each term \(a_n\) of a sequence in terms of the previous \(k\) terms; with initial values, it determines the whole sequence.
\[a_n=f(a_{n-1},a_{n-2},\dots,a_{n-k})\]
More practice problems: recurrence relation
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- ZW-RECURRENCE-RELATION-003 Let a_0=1 and a_n=a_n-1+2n. What is a_5?
- ZW-RECURRENCE-RELATION-004 If a_0=3 and a_n=2a_n-1, which is a formula for a_n?
- ZW-RECURRENCE-RELATION-005 In the Tower of Hanoi, the minimum number of moves satisfies T_n=2T_n-1+1 with T_0=0. What is T_6?
- ZW-RECURRENCE-RELATION-006 How many initial values are needed to determine a sequence with a_n=a_n-1+a_n-2?
- ZW-RECURRENCE-RELATION-007 Let a_n be the number of binary strings of length n with no two consecutive zeros. It satisfies a_n…
- ZW-RECURRENCE-RELATION-008 The sequence with a_0=1,\ a_1=3 and a_n=2a_n-1-a_n-2 satisfies which formula?