Question

A couple decides to buy a house which is currently valued at $318,921.46 on loan. The couple is willing to start paying $200.00 per month and are willing to increase their payment at a rate of 5% every month. How many payments are necessary to pay off the loan amount assuming no deposit was made (answer to the nearest whole number)? What is the value of the final payment that they would make assuming no deposit was made (answer to the nearest whole number)? [Hint: apply geometric sequences].

Answer 1

1. If the first payment is $$200$, next month couple will pay $200 * 1.05$ and the month after this - $200*1.05^2$
2.  Such sequnce of payments is geometric progression.This means that payment made at month number $(n+1)$ is $200*1.05^n$
3. Sum of payments made in the first $(n+1)$ month is equial to:$200+200*1.05+200*1.05^2+ ...+200*1.05^n =\\\\= 200*(1+1.05+1.05^2+ ...+1.05^n)=\\\\=200*(1.05^{n+1}-1)/(1.05-1)=\\\\=200/0.05*(1.05^{n+1}-1)=\\\\=4000*(1.05^{n+1}-1)$
4. This sum should be equial or greater than value of the house $318,921.46.

If $n+1 = 89$, sum of payments will be $303,544.25 which is not enough.

If $n+1 = 90$, sum of payments will be $318,921.460.

This means that $90$ payments are necessary to pay off the loan amount assuming no deposit was made.

Final payment will be $200*1.05^{89} \approx 15377$ dollars assuming no deposit was made.