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Chapter 9: Problem 21

Consider a population that grows linearly following the recursive formula \(P_{N}=P_{N-1}-25,\) with initial population \(P_{0}=578\) (a) Find \(P_{1}, P_{2},\) and \(P_{3}\). (b) Give an explicit formula for \(P_{N}\). (c) Find \(P_{23}\).

Short Answer

Expert verified
The populations \(P_{1}, P_{2}\), and \(P_{3}\) are 553, 528, and 503, respectively. The explicit formula for \(P_{N}\) is \(P_{N}= 578 - 25N\). The population at \(N = 23\) is 3.

Step by step solution

01

Compute P1, P2 and P3

Apply the recursive formula \(P_{N}=P_{N-1}-25\) to compute \(P_{1}, P_{2},\) and \(P_{3}\). Remember, to find the population at time N, you must use the population at time N-1 and subtract 25. For \(P_{1}\), the population at the previous time, \(P_{0}\), is 578, so \(P_{1} = 578 - 25 = 553\). Applying the recursive formula again, \(P_{2} = P_{1} - 25 = 553 - 25 = 528\). Similarly, \(P_{3} = P_{2} - 25 = 528 - 25 = 503\). Therefore, \(P_{1}=553, P_{2} = 528\), and \(P_{3} = 503\).
02

Derive explicit formula for PN

To find an explicit formula for the population at any given time N, notice that the population decreases by 25 at each step. The population at \(P_{N}\) would thus be the initial population, \(P_{0}\) = 578, minus 25 times the number of steps, N. Therefore, the explicit formula for \(P_{N}\) is \(P_{N}= P_{0} - 25N\). When applied, this formula should give the same results for \(P_{1}, P_{2}\), and \(P_{3}\) as we computed using the recursive formula.
03

Compute P23

To find \(P_{23}\), use the explicit formula we derived and substitute N = 23. Thus, \(P_{23} = 578 - 25*23 = 578 - 575 = 3\). Therefore, the population at \(N = 23\) is 3.

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Key Concepts

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

Recursive Formula
A recursive formula is a way to define a sequence or a series where each term depends on one or more previous terms. It essentially tells you how to "build up" from a known starting point. In the case of linear population growth, we often use a recursive formula to predict the population at different times.
For the given exercise, the formula is: \(P_{N}=P_{N-1}-25\). This means that to find the population at time \(N\), you simply take the population from the previous time \(P_{N-1}\) and subtract 25.

Let's break down how it works:
  • Start with a known initial population \(P_{0}\). In this example, \(P_{0}=578\).
  • To find \(P_{1}\), subtract 25 from \(P_{0}\): \(P_{1} = 578 - 25 = 553\).
  • Next, find \(P_{2}\) by subtracting 25 from \(P_{1}\): \(P_{2} = 553 - 25 = 528\).
  • Repeat the process to find \(P_{3}\): \(P_{3} = 528 - 25 = 503\).
The recursive formula is convenient when you need to calculate the next term based on the current one, and it works like a step-by-step strategy.
Explicit Formula
An explicit formula provides a direct way to find any term in a sequence without referring to previous terms. This approach simplifies calculations, especially for finding terms much further in the sequence.
For our linear population growth exercise, the explicit formula derived is: \(P_{N} = P_{0} - 25N\).

This formula says:
  • Start with the initial population \(P_{0}\), which is 578 in our problem.
  • Then subtract \(25 \times N\) to find the population at any point N.
  • \(N\) represents the number of intervals or steps from the starting point.
Using this explicit formula, you can easily calculate \(P_{N}\) for any \(N\) without calculating every preceding term. For example, to find \(P_{23}\), plug into the formula: \(P_{23} = 578 - 25 \times 23 = 3\).
This method is extremely helpful when you need to quickly calculate a term far ahead in the sequence.
Mathematical Modeling
Mathematical modeling is the process of representing real-world situations through mathematical expressions and equations. It's a powerful tool to understand, predict, and design strategies across various fields, including population studies.

When we talk about linear population growth, we use a linear model represented by mathematical formulas. In our exercise, both recursive and explicit formulas are part of this model.
  • Understanding the Situation: The population decreases steadily by a set amount, 25, at each interval. The mathematical model captures this trend.
  • Choosing the Right Formula: Depending on the need, either recursive or explicit formulas can be applied. Use recursive if you are tracing a few steps, or explicit if jumping to far-off values.
  • Predictive Power: With these formulas, you can predict future populations at any step, allowing for better planning and decision-making. For example, knowing \(P_{23} = 3\) indicates a critically low population very soon.
Mathematical modeling helps simplify complex real-world changes into comprehensible mathematical relationships, providing insight into long-term outcomes.

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Most popular questions from this chapter

A manufacturer currently has on hand 387 widgets. During the next 2 years, the manufacturer will be increasing his inventory by 37 widgets per week. (Assume that there are exactly 52 weeks in one year.) Each widget costs 10 cents a week to store. (a) How many widgets will the manufacturer have on hand after 20 weeks? (b) How many widgets will the manufacturer have on hand after \(N\) weeks? (Assume \(N \leq 104\).) (c) What is the cost of storing the original 387 widgets for 2 years (104 weeks)? (d) What is the additional cost of storing the increased inventory of widgets for the next 2 years?

Consider the sequence defined by the recursive formula \(A_{N}=A_{N-1}-2 A_{N-2}\) and starting with \(A_{1}=1, A_{2}=-1\) (a) List the next four terms of the sequence. (b) Find \(A_{8}\).

Show that if \(P_{0}, P_{1}, P_{2}, \ldots\) is an arithmetic sequence, then \(2^{P_{0}}, 2^{P_{1}}, 2^{P_{2}}, \ldots\) must be a geometric sequence.

Crime in Happyville is on the rise. Each year the number of crimes committed increases by \(50 \%\). Assume that there were 200 crimes committed in \(2010,\) and let \(P_{\mathrm{N}}\) denote the number of crimes committed in the year \(2010+N\). (a) Give a recursive description of \(P_{N}\) (b) Give an explicit description of \(P_{N}\) (c) If the trend continues, approximately how many crimes will be committed in Happyville in the year \(2020 ?\)

Consider the sequence \(3,2, \frac{5}{4}, \frac{6}{8}, \frac{7}{16}, \ldots\) (a) List the next two terms of the sequence. (b) If the notation for the sequence is \(A_{1}, A_{2}, A_{3}, \ldots,\) give an explicit formula for \(A_{N}\)
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