Question

Modeling a Geometric Sequence

This activity will help you meet these educational goals:

You will model a geometric sequence with an exponential function, graph the function, and make predictions.

In 2013, scientists began an annual count of butterflies in Elliot Park. That year, they counted 125 butterflies. In subsequent years, the scientists recorded an increase in the park’s butterfly population at a rate of 20% each year. Based on this information, answer the following questions.

Part A
Fill in the table with the values for the population of butterflies. Show how you found each year’s population of butterflies.

Part B
Based on the table of values you created in part A, does a linear or an exponential function best model the butterfly population? Justify your answer by checking for equal differences over equal intervals (linear function) or equal factors over equal intervals (exponential function).
Part C
Question
Plot the population values from the years 2013 through 2016 on the given graph. Include scale and axis labels and a title.

Does this table of values follow an arithmetic or a geometric sequence? Explain.
Part E
Question
Enter the correct answer in the box.
Create a recursive function to model the table of values. In that function, the year 2013 is the first data point, so n = 1 for 2013. This means f(1) = 125 is the first term of the sequence, and r is the constant factor by which the number increases every
year.


Part F
Question
Enter the correct answer in the box.
Rewrite the recursive equation that you got in part E as an explicit equation where the initial value is a and the time elapsed is n years.


Look back at the recursive and explicit functions that you created for this situation in parts E and F. What do the parameters of these equations represent in this specific context?

Answer 1

Answers are below, please ask me if something doesn't make sense.

Explanation:

so first we set up the equation, in an exponential function it has three parts a*b^x where a is the starting amount, b is 1 + the percent increase (or 1 - the percent decrease) and of course x is every time the increase/ decrease happens.

120*1.2^x

Part A

I don't know the table, but just plug in .

Part B

I guess you weren't supposed to know? or just do the math without having the equation. If you ever have an increase by percents or decimals or fractions repeatedly though its an exponential function. Specifcially if they are multiplied. Linear functionshave a term added or subtracted. Or in other words exponential is a*b^x (so b is being repeatedly multiplied) and linear is mx+b so m is continually added. Do you need help with the checking part?

Part C

Graph the points for 0, 1, 2, 3.

Part D

Geometric because there is a common ratio instead of a common difference (repeated multilying vs repeated adding/ subtracting), which is pretty much the definition of a geometric sequence.

Part E

a recursive function would basically just take the parts of the exponential function. a1 = starting point and a(n) = a(n-1)*the common ratio. To make f(1) = 125 we do have to change the function I gave before. Instead of 125*1.2^x it will be 125*1.2^(x-1) . Part C still has the right numbers since we started with 0.

Part F

I gave you the equation, do you know how you would find it though?

The parameters tell us this only works for 2013 and after