Question 1124504
f = p * (1 + r) ^ n


f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.


that's the general formula for the future value of something when compound interest is concerned.


compound interest is a form of exponential growth or decay.


if the car cost 10,000 and decreases 20% in value every 3 years, then:


you can solve as follows:


time period is every 3 years.
interest rate is interest rate per every 3 years.


f = p * (1 + r) ^ n becomes f = 10,000 * (1 - .20) ^ n


simplify this to get f = 10,000 * .80 ^ n


after 3 years, the value of the car is 10,000 * .8 ^ 1 = 8000.
after 6 years, the value of the car is 10,000 * .8 ^ 2 = 6400.
after 9 years, the value of the car is 10,000 * .8 ^ 3 = 5121.536


the average annual growth rate would be calculated as follows:


find the future value after one 3 year period.


assuming the present value is one, the formula would become.


f = 1 * (1 - .2) ^ 1 = 1 * .8 = .8


so, the present value becomes .8 after 3 years.


to find the annual growth rate, you would do the following.


f = p * (1 + r) ^ n becomes .8 = 1 * (1 + r) ^ 3


divide both sides of the equation by 1 to get .8 = (1 + r) ^ 3


take the cube root of both sides of the eqution to get .8 ^ (1/3) = 1 + r


subtract 1 from both sides of the equation to get .8 ^ (1/3) - 1 = r


solve for r to get r = -.0716822333.


that's the annual growth rate.


to confirm, use this rate in the original equation of .8 = 1 * (1 + r) ^ 3


you will get .8 = 1 * (1 - .0716822333) ^ 3


simplify to get .8 = .8


if you graph this equation, it will look like this.


<img src = "http://theo.x10hosting.com/2018/092801.jpg" alt = "$$$" >


from the graph, you can see that the loss in value is 20% every 3 years.


in year 0, the value is 10,000
in year 3, the value is 8000
in year 6, the value is 6400
in year 9, the value is 5120


the every 3 year growth rate is (1 - .2) = .8


the every year growth rate is (1 - .0716822333) = .9283177667