a Let a_n be balance after the n^(th) payment. We are asked to find the balance after the fifth payment, a_5. To do this, we will find a for a_n. Then we will use a spreadsheet.
Recursive Rule
Since the annual interest rate is 9 %=0.09, the monthly interest rate is 0.0912=0.075. This tells us that the balance increases by a factor of 1+0.0075=1.0075 each month, and then the payment of $ 91.37 is subtracted.
ccccc
c Balance after payment
&
=1.0075 *
&
c Balance before payment
&
-
&
Payment
⇓ & & ⇓ & & ⇓
a_n & =1.0075 * & a_(n-1) & - & 91.37
You borrow $2000. This tells us the balance at the beginning of the first month is $2000. Therefore, a_0=2000. Now, we can use a spreadsheet to find the balance after the fifth payment, a_5.
Spreadsheet
Let's enter a_0=5000 in cell A1 and write =Round((1.0075)*(A1)-91.37,2)
in cell A2. When we hit enter, we will get the following.
Next, copy cell A2, highlight cells A3 through A6, and paste.
We get a number 1612.38 in the sixth row. Therefore, a_5=1612.38, and this tells us that the balance after the fifth payment is $1612.38.