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Chapter 10: Problem 4

Find and solve a recurrence relation for the number of ways to park motorcycles and compact cars in a row of \(n\) spaces if each cycle requires one space and each compact needs two. (All cycles are identical in appearance, as are the cars, and we want to use up all the \(n\) spaces.)

Short Answer

Expert verified
The number of ways to park motorcycles and compact cars in a row of \(n\) spaces follows a Fibonacci sequence, where \(F_n = F_{n-1} + F_{n-2}\), and \(F_1=1\), \(F_2=2\).

Step by step solution

01

Identify distinct situations

The last vehicle parked could either be a motorcycle or a compact car. In the case of a motorcycle, there are \(F_{n-1}\) ways to park. If it is a compact car, there are \(F_{n-2}\) ways. Therefore, the total arrangements \(F_n\) can be found by taking the sum of these two situations.
02

Establish the recurrence relation

Considering the previous analysis, it leads to the following recurrence relation: \(F_n = F_{n-1} + F_{n-2}\).
03

Assign the base cases

Base cases of this problem are when \(n=1\), there's only one way to park a bike, \(F_1=1\). Moreover, when \(n=2\), there are two ways to park vehicles, \(F_2=2\) (either two motorcycles or one compact car).
04

Solve the recurrence relation

The recurrence relation is identical to the Fibonacci sequence where each number is the sum of the two preceding ones. Therefore, \(F_n\) is simply the \(n^{th}\) number in the Fibonacci sequence.

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Most popular questions from this chapter

$$ \text { Solve the recurrence relation } a_{n+2}^{2}-5 a_{n+1}^{2}+4 a_{n}^{2}=0, \text { where } n \geq 0 \text { and } a_{0}=4, a_{1}=13 $$

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