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 recursive rule 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.
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c Balance after payment
&
=1.0075 *
&
c Balance before payment
&
-
&
Payment
⇓ & & ⇓ & & ⇓
a_n & =1.0075 * & a_(n-1) & - & 91.37You 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.
b We are asked to find the balance after the last payment. You borrow $2000 for 2 years. Therefore, the 24^(th) payment is the last one. Again, let's use the spreadsheet. Let's enter a_0=2000 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 A26, and paste. Notice that the balance after the first payment is in cell A2. This tells us that the balance after the n^(th) payment is in the (n+1)^(th) row. Therefore, the balance after the usual 24^(th) payment of $ 91.37 will be in cell A25.
The spreadsheet shows that after the usual 24^(th) payment of $ 91.37 the balance is $ 0.02. Therefore, the last payment should be $ 91.37+$ 0.02=$91.39.
