Find the formula

Question

Find a formula for the value, V , of an investment, in dollars, initially worth $11,000 that grows by 11% every 9 years.

Answer 1

The generic equation for exponential growth is V = V₀(1 + r)^t where V is the future amount, V₀ is the initial amount, r is the rate, and t is the time but it assumes that the units of the rate (r) and time (t) are the same. So when we talk about the generic equation for compound interest as V = V₀(1 + r/n)^nt where n is the number of compounding periods, this is just a process to use an annual rate and adjust it to be the same units as the exponent. For example, for interest compounded monthly I would divide the annual rate by 12 turning it into a monthly rate and then I would multiply the number of years by 12 turning the exponent into months.

In this case, it's a little weird though because the interest rate is not per year but per every 9 years.

So we could just say V = 11000(1 + 0.11)^x where x is 9 year periods. So, in other words, for 9 years, we would use x = 1, for 18 years x = 2, for 27 years x = 3, etc. But that's a little weird so another possibility is that we convert the time into 9 year blocks within the equation by just having t be in years and dividing the number of years by 9 making the equation V = 11000(1 + 0.11)^t/9 where t is in years. That way, when t = 9, the exponent will be 9/9 or 1 or when t = 18, the exponent will be 18/9 or 2.

That seems OK until we consider years in between. So, technically, if you only get the 11% at the end of 9 years, that won't accurately predict the amount for 8 years.

The only way I can think to fix this is to use the Greatest Integer function. The greatest integer function (GIF) returns the integer just smaller or equal to the number (so the GIF of 2.6 is 2). Sometimes the GIF notation is [x] and I've also seen [[x]] so I'll just call it GIF(x). Using the GIF, we can say the growth function is V = 11000(1 + 0.11)^GIF(t/9) meaning you don't get the interest until the 9 years is up.

So, I guess it depends on what they mean by it "grows by 11% every 9 years". If they mean you don't get the interest until the 9 years is up, then it is: V = 11000(1 + 0.11)^GIF(t/9) but if they mean it averages 11% every 9 years then you could get by with V = 11000(1 + 0.11)^t/9

Answer 2

V = 11,000*(1.11)^(t/9), where t is time in years.this formula is a function the outputs the values of required by the question, V at t =0 is 11000 and V at 9 years is 11000+11% and that amount grows by 11% at 18 years, and so on.....