Given that the 3% interest per year is compounded monthly and that the person saving up adds $200 to the account each month:
If for each integer ($n$) greater than 0, $A_n$ is the amount the account has at the end of $n$ months, what is the recurrence relation relating $A_k$ to $A_{k-1}$ ?
At the beginning of month $k$, bank deposits interest accrued based on the balance of the previous month $A_{k-1}$. The interest accrued equals $I_{k} = r_{m} A_{k-1}$ for some fraction $r_{m}$. In addition, the account holder deposits the amount $D_{k}$. Hence
$$ A_{k} = A_{k-1} + r_{m} A_{k-1} + D_{k} $$
To determine $r_{m}$, one needs to decipher "3% annual interest, compounded monthly". It means that monthly interest is the early interest, divided by the number of months in a year: $$ r_m = \frac{r_y}{12} = \frac{1}{400} $$
