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)2Which of the following describes what is happening to the insect population for the first five years?

Starting with a₁ = 8, the recurrence gives a₂ = 2.9(8) − 0.2(8²) = 10.4, so the population increases. The next values are approximately a₃ = 8.528, a₄ = 10.186, and a₅ = 8.790, so it decreases, increases, then decreases. Therefore, the population alternates between increasing and decreasing, as choice D states.

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