Question

The growth of a local raccoon population approximates a geometric sequence where f(n) is the number of raccoons in a given year and n is the year.After 6 years there are 45 raccoons and after 8 years there are 71 raccoons.Write an explicit rule in function notation that models the local raccoon population in the terms of the number of years.

Answer 1

f(n) = 14.3908 * 1.2561^(n-1)

Explanation:

A geometric sequence can be defined by:

f(n) = a*r^(n-1), where 'a' is the inicial population, and 'r' is the ratio the population increases each year

If we have 45 raccoons after 6 years and 71 raccoons after 8 years, we can use these values in the equation to find the values of 'a' and 'r':

for n=6, f(n) = 45:

45 = a*r^5

for n=8, f(n) = 71:

71 = a*r^7

dividing the second equation by the first, we have:

r^2 = 71/45 = 1.5778

r = 1.2561

Now, applying this value of 'r' in the first equation, we find 'a':

45 = a*1.2561^5

a = 45/3.1270 = 14.3908

So, the function that models the local raccoon population 'f(n)' in the terms of the number of years 'n' is:

f(n) = 14.3908 * 1.2561^(n-1)