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Chapter 9: Problem 20

Consider a population that grows linearly following the recursive formula \(P_{N}=P_{N-1}+23,\) with initial population \(P_{0}=57\) (a) Find \(P_{1}, P_{2},\) and \(P_{3}\) (b) Give an explicit formula for \(P_{N}\). (c) Find \(P_{200}\)

Short Answer

Expert verified
\(P_{1}=80, P_{2}=103, P_{3}=126, P_{N}=57+23*N, P_{200}=4657\)

Step by step solution

01

Compute \(P_{1}, P_{2},\) and \(P_{3}\)

Starting from the initial population \(P_{0}=57, P_{1}\) can be found by substituting \(N=1\) in the given formula: \(P_{1}=P_{1-1}+23=P_{0}+23=57+23=80\). Likewise, \(P_{2}=P_{2-1}+23=P_{1}+23=80+23=103\) and \(P_{3}=P_{3-1}+23=P_{2}+23=103+23=126\)
02

Formulate an explicit formula for \(P_{N}\)

We can construct an explicit formula based on the recursive relationship. Notice that each term is 23 more than the previous term, which is a characteristic of arithmetic sequences. The explicit formula for an arithmetic sequence is \(P_{N}=P_{0}+d*N\), where \(d\) is the common difference, and \(N\) is the term number. Substituting into this equation with \(P_{0}=57\) and \(d=23\), we get \(P_{N}=57+23*N\)
03

Find \(P_{200}\)

To find \(P_{200}\), substitute \(N=200\) in the explicit formula: \(P_{200}=57+23*200=4657\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Growth Models
Linear growth models describe processes that increase by the same amount in each time period. These models are straightforward and predictable, making them useful in various fields like population studies, finance, and more.
For instance, if a population grows by 23 individuals every year, like in our example, then each year adds a constant increment to the population total. This simplicity allows us easily to observe and predict future numbers.
  • Predictability: With linear growth, we know what to expect each period. The changes are consistent.
  • Simplicity: Calculations are simple, so linear models are often the first choice for basic growth modeling.
This model is less realistic for long-term predictions since real-world situations can vary, but it's a perfect fit for understanding basic incremental changes.
Arithmetic Sequences
An arithmetic sequence is a list of numbers where each term after the first is generated by adding a constant, known as the common difference, to the previous term.
In our example, the population increases by 23 each year, creating an arithmetic sequence: 57, 80, 103, 126, ...
  • Common Difference: Here, it's 23, showing the amount added each time.
  • Formula: The nth term can be written as \( a_n = a_1 + (n-1) \times d \), where \( a_1 \) is the first term.
This helps in quickly finding any term in the sequence without computing all previous ones, which is especially handy when dealing with large numbers or indices.
Explicit Formula Construction
The explicit formula allows us to directly calculate any term in a sequence without relying on its predecessors, unlike recursive formulas which build off the previous one.
To form an explicit formula, observe the arithmetic pattern, noting initial value and common difference. For our problem, the explicit formula is:
  • Initial Value \( P_0 = 57 \)
  • Common Difference \( d = 23 \)
Resulting in: \( P_N = 57 + 23 \times N \)
This formula is powerful as it allows calculation of any term like \( P_{200} \) instantly, which in our case was computed as 4657. This efficiency is crucial in scenarios needing quick or remote access to specific data points.

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Most popular questions from this chapter

Consider the sequence \(2,3,5,9,17,33, \ldots\) (a) List the next two terms of the sequence. (b) Assuming the sequence is denoted by \(A_{1}, A_{2}, A_{3}, \ldots\), give an explicit formula for \(A_{N}\). (c) Assuming the sequence is denoted by \(P_{0}, P_{1}, P_{2}, \ldots\). give an explicit formula for \(P_{N}\).

Suppose that you are in charge of stocking a lake with a certain type of alligator with a growth parameter \(r=0.8\). Assuming that the population of alligators grows according to the logistic growth model, is it possible for you to stock the lake so that the alligator population is constant? Explain.

The purpose of this exercise is to understand why we assume that, under the logistic growth model, the growth parameter \(r\) is between 0 and 4 (a) What does the logistic equation give for \(p_{N+1}\) if \(p_{N}=0.5\) and \(r>4 ?\) Is this a problem? (b) What does the logistic equation predict for future generations if \(p_{N}=0.5\) and \(r=4 ?\) (c) If \(0 \leq p \leq 1\), what is the largest possible value of \((1-p) p ?\) (d) Explain why, if \(0

Consider the sequence \(0,1,3,6,10,15,21, \ldots\) (a) List the next two terms of the sequence. (b) Assuming the sequence is denoted by \(A_{1}, A_{2}, A_{3}, \ldots\) give an explicit formula for \(A_{N}\). (c) Assuming the sequence is denoted by \(P_{0}, P_{1}, P_{2}, \ldots\), give an explicit formula for \(P_{N}\).

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