Solve each problem. Suppose an insect population density, in...

Solve each problem. Suppose an insect population density, in thousands per acre, during year $n$ can be modeled by the recursively defined sequence $$\begin{aligned} &a_{1}=8\\ &a_{n}=2.9 a_{n-1}-0.2 a_{n-1}^{2}, \quad \text { for } n>1 \end{aligned}$$ (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.

Solve each problem.
Suppose an insect population density, in thousands per acre, during year $n$ can be modeled by the recursively defined sequence
$$\begin{aligned}
&a_{1}=8\\
&a_{n}=2.9 a_{n-1}-0.2 a_{n-1}^{2}, \quad \text { for } n>1
\end{aligned}$$
(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.

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Key Concepts

Recursive Sequences

A recursive sequence is defined by an initial term or terms and a rule that expresses each subsequent term as a function of the preceding term(s). Understanding how to apply the recursive rule repeatedly is crucial for finding specific terms in the sequence.

Iterative Computation

It is important to grasp the method of iterative computation, where one starts from a known initial condition and then applies the recursive relationship repeatedly to generate future terms. This method is commonly used in problems involving step-by-step progression.

Discrete Dynamical Systems

The sequence given can be viewed as a discrete dynamical system, where the evolution of the system is modeled with a difference equation. Such systems are pivotal in studying how populations or other quantities change over time under given rules.

Graphical Analysis

Graphing the terms of a sequence allows for visualization of the behavior of the system over time. By plotting a series of computed values against the time or iteration variable, one can interpret trends, convergence, or other dynamical properties of the sequence.

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