College Algebra, Chapter 9, 9.1, Section 9.1, Problem 78

Suppose that a biologist is trying to find the optimal salt concentration for the growth of a certain species of mollusk.
She begins with a brine solution that has $\displaystyle 4 \frac{\text{g}}{\text{L}}$ of salt and increases the concentration $10\%$ every day.
Let $C_0$ denote the initial concentration and $C_n$ the concentration after $n$ days.
a.) Find a recursive definition of $C_n$
b.) Find the salt concentration after 8 days.

a.) If the initial concentration is 4, then $C_0 = 4$. So, $C_n = 0.10 C_{(n-1)} + 4$

b.) After 8 days, the salt concentration will be

$
\begin{equation}
\begin{aligned}
C_1 &= 0.10 C_{(1-1)} + 4 &&& C_2 &= 0.10 C_{(2-1)} + 4\\
\\
&= 0.10 C_0 + 4 &&& &= 0.10 C_1 + 4\\
\\
&= 0.10 (4) +4 &&& &= 4.44\\
\\
&= 4.40 &&& C_3 &= 0.10 C_{(3-1)} + 4\\
\\
C_4 &= 0.10 C_{(4-1)} + 4 &&& &= 0.10 C_2 + 4\\
\\
&= 0.10 C_3 + 4 &&& &= 4.444\\
\\
&= 4.4444 &&& C_5 &= 0.10 C_{(5-1)} + 4\\
\\
C_6 &= 0.10 C_{(6-1)} + 4 &&& &= 0.10 C_4 + 4\\
\\
&= 0.10 C_5 + 4 &&& &= 4.44444\\
\\
&= 4.444444 &&& C_7 &= 0.10 C_{(7-1)} + 4\\
\\
C_8 &= 0.10 C_{(8-1)} + 4 &&& &= 0.10C_6 + 4\\
\\
&= 0.10 C_7 + 4 &&& &=4.4444444\\
\\
&= 4.44444444
\end{aligned}
\end{equation}
$

Therefore, the salt concentration after 8 days is $\displaystyle 4.44444444 \frac{\text{g}}{\text{L}}$

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