College Algebra, Chapter 9, 9.1, Section 9.1, Problem 76

The median price of a house in Orange County increases by about $6\%$ per year. In 2002, the median price was $240,000$. Let $P_n$ be the median price $n$ years after $2002$.
a.) Find a formula for the sequence $P_n$
b.) Find the expected median price in $2010$.

a.) If we let the initial median price be $\$240,000$, then $P_n = 0.06 [P_{(n-1)}] + 240000$

b.) The median price in $2010$ where $n = 8$ is


$
\begin{equation}
\begin{aligned}
P_1 &= 0.06 [P_{(n-1)}] + 240000 &&& P_4 &= 0.06 P_3 + 240000\\
\\
&= 0.06 P_0 + 240000 &&& &= 255318.9504\\
\\
&= 0.06 (240000) + 240000 &&& P_5 &= 0.06 P_4 + 240000 \\
\\
&= 254,400 &&& &= 255319.137\\
\\
P_2 &= 0.06 P_1 + 240000 &&& P_6 &= 0.06 P_5 + 240000\\
\\
&= 0.06(254400) + 240000 &&& &= 255319.1482\\
\\
&= 255264 &&& P_7 &= 0.06 P_6 + 240000\\
\\
P_3 &= 0.06 P_2 + 240000 &&& &=255319.1489\\
\\
&= 0.06(255264) + 240000 &&& P_8 &= 0.06 P_7 + 240000\\
\\
&= 255315.84 &&& &= 255319.1489
\end{aligned}
\end{equation}
$

Therefore, the median price of the house in the year $2010$ will be $\$255,319.1489$