Question

In 2008, a 60 Month loan at 7.3% on a new Ford Mustang (Financing $27,840) will result in a monthly payment of $555.21. As a rule of thumb, a car depreciates in value by 15% every year.

After making 60 payments (5 years), How much money has been paid out?

Explain why we can say that the 15% depreciation of the value of the car is an example of a geometric sequence.

Answer 1

1.
Total money paid out=Monthly payments*Number of months=555.21*60
=33312.6

2.
Depreciation in Year 1=27840*15%

Depreciation in Year 2=(27840-27840*15%)*15%=27840*(1-15%)*15%=27840*15%*(1-15%)^1

Depreciation in Year 2=(27840-27840*(15%-15%*15%+15%))*15%=27840*(1-15%-15%+15%*15%)*15%=27840*15%*(1-2*15%+15%^2)=27840*15%*(1-15%)^2

Thus depreciation in year n=27840*15%*(1-15%)^n=initial value*depreciation rate*(1-depreciation rate)^n

This looks like a geometric sequence

It is a geomteric sequence because the depreciation is calculated on the book value (which is calculated after subtracting depreciation from initial value)