Question

Suppose that an insect population’s density, in thousands per acre, during year n, can be modeled by the recursive formula:
a1 = 8
an = 2.9an – 1 – 0.2(an – 1)2
Which of the following describes what is happening to the insect population for the first five years?

The population density decreases each year.

The population density increases each year.

The population density remains constant.

The population alternates between increasing and decreasing.

Answer 1

Compute the terms:

- a1 = 8
- a2 = 2.9(8) − 0.2(8)^2 = 10.4 (increase)
- a3 = 2.9(10.4) − 0.2(10.4)^2 ≈ 8.528 (decrease)
- a4 = 2.9(8.528) − 0.2(8.528)^2 ≈ 10.186 (increase)
- a5 ≈ 8.789 (decrease)

The population alternates between increasing and decreasing over the first five years.