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25 December, 05:47

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.

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  1. 25 December, 06:16
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    f (n) = 14.3908 * 1.2561^ (n-1)

    Step-by-step 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)
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