College Algebra, Chapter 9, 9.1, Section 9.1, Problem 74

Suppose that Margarita borrows $\$10,000$ from her uncle and agrees to repay it in monthly installments of $\$200$. Her uncle charges $0.5\%$ interest per month on the balance.
a.) Show that her balance $A_n$ in the $n$th month is given recursively by $A_n = 10,000$ and
$A_n = 1.005 A_{n-1} - 200$
Find her balance after six months.

a.) Margarita's balance after 1 month is

$
\begin{equation}
\begin{aligned}
A_1 &= 1.005 A_{(1-1)} - 200\\
\\
A_1 &= 1.005 A_0 - 200\\
\\
A_1 &= 1.005 (10,000) - 200\\
\\
A_1 &= \$ 9850
\end{aligned}
\end{equation}
$


Thus, the given is an example of a recursively defined sequence. Since it allows us to find the $n$th term if we know the preceding term.

b.) If $A_1 = 9850$, then

$
\begin{equation}
\begin{aligned}
A_2 &= 1.005 A_1 - 200 &&& A_4 &= 1.005 A_3 - 200\\
\\
A_2 &= 1.005 (9850) - 200 &&& A_4 &= 1.005 (9547.75) - 200\\
\\
A_2 &= \$ 9,699.25 &&& A_4 &= \$9,395.48\\
\\
A_3 &= 1.005 A_3 - 200 &&& A_5 &= 1.005 A_4 - 200\\
\\
A_3 &= 1.005 (9699.25)-200 &&& A_5 &= 1.005 (9395.48) - 200\\
\\
A_3 &= \$ 9,547. 75 &&& A_5 &= \$9,242.46
\end{aligned}
\end{equation}
$

Then the balance after 6 months is
$A_6 = 1.005 A_5 - 200$
$A_6 = 1.005 (9242.46) - 200$
$A_6 = \$9088.67$