Explanation:
The rate of increase yearly is
[tex]\begin{gathered} r=69\% \\ r=\frac{69}{100}=0.69 \end{gathered}[/tex]
The number of lionfish in the first year is given beow as
[tex]N_0=9000[/tex]
Part A:
To figure out the explicit formula of the number of fish after n years will be represented using the formula below
[tex]P(n)=N_0(1+r)^n[/tex]
By substituting the formula, we will have
[tex]\begin{gathered} P(n)=N_{0}(1+r)^{n} \\ P(n)=9000(1+0.69)^n \\ P(n)=9000(1.69)^n \end{gathered}[/tex]
Hence,
The final answer is
[tex]f(n)=9,000(1.69)^n[/tex]
Part B:
to figure out the amoutn of lionfish after 6 years, we wwill substitute the value of n=6
[tex]\begin{gathered} P(n)=9,000(1.69)^{n} \\ f(6)=9000(1.69)^6 \\ f(6)=209,683 \end{gathered}[/tex]
Hence,
The final answer is
[tex]\Rightarrow209,683[/tex]
Part C:
To figure out the recursive equation of f(n), we will use the formula below
From the question the common difference is
[tex]d=-1400[/tex]
Hence,
The recursive formula will be
[tex]f(n)=f_{n-1}-1400,f_0=9000[/tex]
