An invasive species of rodent is spreading across the south. The initial population was estimated at 400 individuals in 2010, but the population has been growing by 58% every year. How many of these rodents can we expect by the year 2025 if the rate they are spreading remains the same?

1,871
871
81,871
381,871

Question

An invasive species of rodent is spreading across the south. The initial population was estimated at 400 individuals in 2010, but the population has been growing by 58% every year. How many of these rodents can we expect by the year 2025 if the rate they are spreading remains the same?

1,871
871
81,871
381,871

holsteremission · Answer

If $y_n$ is the population at the $n$th year, and the population grows by a factor of $58\%$ each year, then you can model the population recursively by
$$\begin{cases}
y_0=400\[1ex]
y_n=1.58y_{n-1}&\text{for }n\ge1
\end{cases}$$where $n=0$ corresponds to the year $2010$.

You can solve for $y_n$ inductively:
$$y_n=1.58y_{n-1}=1.58^2y_{n-2}=1.58^3y_{n-3}=\cdots=1.58^ny_0$$The population in $2025$ would be given by $y_{25}$.