Problem 84 - A Graphical Approach to College Algebra

Question

Solve each problem. Suppose an insect population density in thousands per acre during year \(n\) can be modeled by the following recursively defined sequence. $$\begin{aligned}&a_{1}=8\\\&a_{n}=2.9 a_{n-1}-0.2 a_{n-1}^{2}, \text { for } n>1\end{aligned}$$ (a) Find the population for \(n=1,2,3\). (b) Graph the sequence for \(n=1,2,3, \ldots, 20 .\) Use the window \([0,21]\) by \([0,14] .\) Interpret the graph.

Step-by-Step Solution

Answer

(a) 8, 10.4, 8.528. (b) Population rises initially, then stabilizes.

Step 1: Understanding the Recurrent Formula

The sequence is given by two parts: the initial value \( a_1 = 8 \) and a recursive formula \( a_n = 2.9a_{n-1} - 0.2a_{n-1}^2 \) for \( n > 1 \). Here, each term depends on the previous term in the sequence.

Step 2: Calculate Population for n=1

From the initial condition, we know that \( a_1 = 8 \). This is given directly by the problem, so we have the first term or the population for year 1 as `8 thousand insects per acre`.

Step 3: Calculate Population for n=2

To find \( a_2 \), use the recursive formula: \( a_2 = 2.9 \times 8 - 0.2 \times 8^2 \). Substitute and solve:\[a_2 = 2.9 \times 8 - 0.2 \times 64 = 23.2 - 12.8 = 10.4\]This means at the second year, the population is `10.4 thousand insects per acre`.

Step 4: Calculate Population for n=3

Now, calculate \( a_3 \) using \( a_2 = 10.4 \): \( a_3 = 2.9 \times 10.4 - 0.2 \times 10.4^2 \).Compute each part:\[ 2.9 \times 10.4 = 30.16 \]\[ 0.2 \times 10.4^2 = 0.2 \times 108.16 = 21.632 \]\[ a_3 = 30.16 - 21.632 = 8.528 \]Thus, the population for the third year is `8.528 thousand insects per acre`.

Step 5: Set Up Graphing Conditions

You need to graph the sequence values \( a_n \) from \( n=1 \) to \( n=20 \). To do this, set the graph window with x-values from 0 to 21 (for years) and y-values from 0 to 14 (for population in thousands) as given in the problem statement.

Step 6: Interpret the Graph

After plotting the sequence \( a_n \), you'll observe that the population initially increases but stabilizes or fluctuates as \( n \) becomes larger. This is typical of sequences where growth is limited by a decreasing factor, such as competition limiting further exponential growth.

Key Concepts

Population Modeling

Population modeling is a mathematical approach used to predict and understand the changes in populations over time. In our example, the insect population is modeled using a recursive sequence. This model takes into account factors that affect population growth and decay.

• The initial population, given as 8, represents the starting point of our model.
• The recursive formula describes how the population in the next year depends on the population from the previous year. This captures elements like growth rate and limitation factors.

An important aspect of population models, like the one presented, is to understand how different factors either contribute to growth or lead to decline. The term \(2.9a_{n-1}\) represents growth, while \(0.2a_{n-1}^2\) reflects limitations such as resource competition or environmental carrying capacity. These models are valuable in predicting long-term trends and are widely used in ecology, conservation, and resource management.

Graphical Interpretation

Graphical interpretation involves analyzing the graphical representation of the sequence to glean insights about the population's behavior over time. By examining the graph, we can observe trends and patterns that would be less obvious from mere calculations. When we graph the sequence for years 1 through 20, the x-axis represents the years, while the y-axis represents the population in thousands. Initially, the graph shows an increase, which slows and may fluctuate as time progresses.

• The initial increase is often due to exponential growth, seen when populations have abundant resources.
• The stabilization or fluctuation reflects how real-world factors limit growth, ensuring populations do not grow indefinitely.

Interpreting these trends can tell us about the population's adaptability to changes and its ability to reach equilibrium over time. This insight is crucial for making predictions and informed decisions about managing populations in their environments.

Sequence Graphing

Sequence graphing is the visual plotting of terms from a recursive or explicit sequence on a coordinate system. It helps in understanding the progression and dynamics of sequences over time. To graph the sequence in the given problem, set the range for years from 0 to 21 and the population from 0 to 14. Each point plotted represents the insect population for a specific year.

• The initial points of the sequence tend to show a rapid increase as seen from year 1 to year 3.
• As the sequence progresses, the points on the graph may start to level off, reflecting constraints included in the recursive model.

By graphing, we can visually grasp how each year's population impacts the next, providing a clear view of both growth patterns and constraints. This graphical tool simplifies interpretation by visually highlighting trends and behaviors that numbers alone might obscure. Sequence graphing is a pivotal skill in population modeling, offering a means to both explore and explain complex biological and mathematical phenomena.