Answer
$155,797$
Work Step by Step
1. Familiarize yourself with the problem.
In one year, the population will be,
$100,000+0.03\left( 100,000 \right)=\left( 1.03 \right)100,000$.
In two years, the population will be,
$\left( 1.03 \right)100,000+\left( 0.03 \right)\left( 1.03 \right)\left( 100,000 \right)={{\left( 1.03 \right)}^{2}}100,000$
The population forms a geometry series:
$100,000,\left( 1.03 \right)100,000,{{\left( 1.03 \right)}^{2}}100,000,\ldots $
Thus, the population at 15 years will be the 16th term of the geometry series.
2. Translate the problem into an equation,
$\begin{align}
& {{a}_{1}}=100,000 \\
& r=\frac{\left( 1.03 \right)100,000}{100,000} \\
& =1.03 \\
& n=16
\end{align}$
$\begin{align}
& {{a}_{n}}={{a}_{1}}{{r}^{n-1}} \\
& {{a}_{16}}=\left( 100,000 \right){{\left( 1.03 \right)}^{16-1}}
\end{align}$
3. Carry out the mathematical operations to solve the equation.
Substitute $n=16,{{a}_{1}}=100,000$ and $r=1.03$,
$\begin{align}
& {{a}_{n}}={{a}_{1}}{{r}^{n-1}} \\
& =\left( 100,000 \right){{\left( 1.03 \right)}^{16-1}} \\
& =\left( 100,000 \right){{\left( 1.03 \right)}^{15}} \\
& =155,797
\end{align}$
Thus, the approximate population in 15 years is $155,797$.
