Let a_n be balance after the n^(th) payment. We are asked to find how long it takes to pay back the loan. 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 7.5 %=0.075, the monthly interest rate is 0.075/12= 0.00625. This tells us that the balance increases by a factor of 1+ 0.00625=1.00625 each month. At the end of the month, the payment of $ 1048.82 is subtracted.
ccccc
c Balance after payment
&
=1.00625 *
&
c Balance before payment
&
-
&
Payment
⇓ & & ⇓ & & ⇓
a_n & =1.00625 * & a_(n-1) & - & 1048.82
You borrow $150 000 for 30 years. This tells us the balance at the beginning of the first month is $150 000. Therefore, a_0=150 000. We are asked to find the amount of the last, 360^(th), payment. To do this, we will use a spreadsheet.
Spreadsheet
Let's enter a_0=150 000 in cell A1 and write =Round((1.00625)*(A1)-1048.82,2) in cell A2. When we hit enter we will get the following.
Next, copy cell A2, highlight cells A3 through A361, 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 360^(th) payment of $ 1048.82 will be in cell A361.
The spreadsheet in the 361^(th) row shows that after the usual 360^(th) payment of $ 1048.82 the balance is $ 2.21. Therefore, the last payment should be $ 1048.82+$ 2.21=$1051.03.
