College Algebra, Chapter 9, Review Exercises, Section Review Exercises, Problem 30

A teacher makes $\$35,000$ in his first year, and gets a $\$1,200$ raise each year.
a.) What is his salary $A_n$ in his $n$th year at this school?
b.) Find his salary in his eighth year at this this school, and compare it to the salary of the teacher in her eighth year that makes $\$32,000$ in her first
year and gets a $5\%$ raise each year.

a.) Since the raise of the teacher is fixed at $\$1,200$ each year, then the salary of the teacher can be represented by the criteria of
$A_n = 35,000 + (n - 1)(1,200)$

So the salary of the male teacher in his eight year is

$
\begin{equation}
\begin{aligned}
A_8 &= 35,000 + (8 - 1) ( 1,200)\\
\\
A_8 &= \$43,400
\end{aligned}
\end{equation}
$


In the case of the female teacher, her salary for the first year is $32,000$ while her salary for the year is $32,000 + (0.05)32,000$. Notice that,
the salary of the female teacher can be represented by the geometric sequence of

$
\begin{equation}
\begin{aligned}
A_n &= a(r)^{n-1}\\
\\
A_n &= 32,000 \left( \frac{32,000+(0.05)(32,000)}{32,000} \right)^{n-1}\\
\\
A_n &= 32,000 \left( \frac{1.05 \cancel{(32,000)}}{\cancel{32,000}} \right)^{n-1}\\
\\
A_n &= 32,000 (1.05)^{n-1}
\end{aligned}
\end{equation}
$

Therefore, the salary of the female teacher on her eighth year is

$
\begin{equation}
\begin{aligned}
A_8 &= 32,000(1.05)^{8-1}\\
\\
A_8 &= \$45,027.21
\end{aligned}
\end{equation}
$

It shows that the salary of the male teacher is initially higher than the salary of the female teacher. But, eventually the salary of the female
teacher will be higher since the increase in the salary of the male teacher is fixed.