91Ó°ÊÓ

Jodi and Jeff borrowed $120,000 at 6.5% per annum compounded monthly for 30 years to purchase a home. Their monthly payment is determined to be $758.48.

Given that $A=758.48,r=0.065,n=12,p_0=120,000$

The recursive formula can be found by substituting the values in the equation role="math" localid="1648522221518" $p_n=\left(1+\frac{r}{n}\right)p_{n-1}-758.48$

role="math" localid="1648522347330" $$\begin{gathered} p_n=\left(1+\frac{r}{n}\right)p_{n-1}-758.48 \\ {p}_n=\left(1+\frac{0.065}{12}\right)p_{n11}-758.48 \\ {p}_n=1.0054p_{n-1}-758.48 \end{gathered}$$

To calculate Jodi and Jeff’s balance after the first payment we have to evaluate the recursive formula for $n=1$

$$\begin{gathered} p_1=1.0054p_{1-1}-758.48 \\ {p}_1=1.0054p_0-758.48 \\ {p}_1=1.0054×120,000-758.48 \\ {p}_1=119889.52 \end{gathered}$$

The table showing Jodi and Jeff’s balance after each monthly payment can be created by evaluating the recursive formula for different values of $n$.

The table showing Jodi and Jeff’s balance is

From the above table, we can conclude that, $p_n$is less than $100,000 for $n=127$

Therefore the balance will be less than $100,000 after 127 months.

The table showing Jodi and Jeff’s balance is

From the table, we can conclude that $p_n$is less than $0$for $n=358$

Therefore the balance will be paid off after 358 months.

Jodi and Jeff’s interest expenses can be calculated by subtracting the borrowed amount from the total of all payments.

From the table of the previous part, we can say that the balance falls $0 after the 358th payment

The payment of $758.48 was made 357 times

The last payment paid by Jodi and Jeff can be calculated by subtracting $-213.3 from $758.48

$758.48-213.3=545.18$

The total of all payments is the sum of $758.48 payments made for 358 months and the sum of the last payment

$358×758.48+545.18=271322.54$

The interest expense is

$271322.54-120,000=151322.54$

If Jodi and Jeff pay an extra $100 then the interest expense will be significantly lower, therefore it is worthwhile for Jodi and Jeff to pay the additional $100