Using Algebra and Discrete Mathematics to Investigate Population Changes in a Trout Pond Part Four - Symbolic Analysis

	Make a Conjecture	Numerical Analysis	Graphical Analysis	Symbolic Analysis

In the previous parts of this i-Math Investigation you carried out a graphical and numerical analysis of the trout population problem. You also worked with some equations. It would be useful to find additional equations or formulas that describe this situation. Then you could reason about those equations or formulas to help you better understand and analyze this situation. Reasoning About the Recursion Equation A(n) = 0.8 A(n-1) + 1000 is one of the equations that models the original trout population problem. (This type of equation can be called a "recursion equation" because it involves recursion, whereby the present population is determined from the previous population.) Try to find a way to reason about this equation to determine the long-term population. Explain your reasoning. Use this HINT if you need it. Explain why solving x = 0.8 x + 1000 will give the long-term population. How is this seen in the stairstep graph you generated in the graphical analysis in Part 3? Use this HINT if you need it.

Finding an Explicit Formula Try to find another equation that represents this situation - an equation that has this form: A(n) = "some expression involving n, and not A(n-1)" Such an equation is sometimes called an "explicit formula", because it gives the population explicitly in terms of n. Use this sequence of HINTS if you need it. Reasoning About the Explicit Formula Use the explicit formula to analyze the trout population problem in new ways, as follows: Use the explicit formula to find the population after 10 years. Explain how this method is different from using the recursion equation to find the population after 10 years. Use the explicit formula to find the long-term population. Explain your method and compare to the result you got in previous parts of this i-Math Investigation. Consider the general form of the recursion equation in this situation: A(n) = r A(n-1) + b, where 0 < r < 1 What is the explicit formula for A(n) in this general case? SOLUTION Reason about the general explicit formula above to find a general formula for the long-term population in situations like this. Use this general formula to find the long-term population in the original trout population problem. Use this HINT if you need it.

Completing this i-Math Write a paragraph summarizing how tables of values, graphs, and equations can be used to analyze the changing trout population. Make a Conjecture Numerical Analysis Graphical Analysis Symbolic Analysis References and Credits

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