Answer
(a) See explanations.
(b) $9088.675$ dollars.
Work Step by Step
(a) Step 1. The starting balance is $A_0=10000$.
Step 2. With a $0.5\%$ interest per month and a $200$ payment, the balance at the end of the first month becomes $A_1=1.005A_0-200$.
Step 3. The second month will repeat the process of step 2 to give $A_2=1.005A_1-200$ ......
Step 4. Repeat this process for $n-$months, we get $A_n=1.005A_{n-1}-200$
(b) After 6 month, we have $A_6=1.005A_5-200=1.005(1.005A_4-200)-200=...=9088.675$ dollars.
The calculation of this series up to 10 month is shown in the figure.
