Question

A native wolf species has been reintroduced into a national forest. Originally 200 wolves were transplanted. After 3 years, the population had grown to 270 wolves. If the population grows exponentially,

Write a recursive formula for the number of wolves

Answer 1

A recursive formula for an exponentially growing wolf population, where initially there were 200 wolves and 270 after 3 years, is Pn = Pn-1 × (270/200)^(1/3) for n ≥ 1, starting with P0 = 200.

Step-by-step explanation:

To write a recursive formula for the number of wolves in the population which grows exponentially, we observe that originally 200 wolves were transplanted and after 3 years the population grew to 270 wolves. A recursive formula will describe the number of wolves in year n based on the number of wolves in year n-1. Let's define Pn as the population of wolves in year n, and assume the growth rate remains constant.

We know that P0 = 200 (initial population) and from year 0 to year 3 the population grew to 270 wolves. The growth factor over 3 years is 270/200 which implies an annual growth factor of (270/200)^(1/3). Therefore, the recursive formula can be written as:

Pn = Pn-1 × (270/200)^(1/3) for n ≥ 1, where P0 = 200

This formula allows us to calculate the population for any subsequent year n by multiplying the previous year's population by the growth factor.