Exponential growth

[sidius]

This is problem 07G-36 from the 2011 Stated Problem Study Guide:

Kaitlynn makes four equal annual payments to a bank that pays 6% annual interest. At the end of the fourth year, she has earned $500. How much was one of the four payments?

The answer according to the answer sheet is $784.81. Through working backwards, I see how this answer is achieved:

monthly payment = 500 / [(((1+.06)^5 - 1)/.06) - 5] = $784.81

My problem is that I have no idea why it needs to be "^5" and "-5" considering it was four annual payments over four years.
What am I missing? How would the solution change if the problem said she "made six equal annual payments...how much was one of the six annual payments?"

[SumitG]

I'm not really sure if there is an intuitive combinatorial argument which gives your closed form solution to this problem...I only know how to find it algebraically. So like, I'm not exactly sure "why" intuitively you would use [unparseable or potentially dangerous latex formula] instead of [unparseable or potentially dangerous latex formula], or explain the [unparseable or potentially dangerous latex formula], but I can show you mathematically that it does work out that way, and how manipulation of your original data will affect your solution.

Consider that you have a rate of interest [unparseable or potentially dangerous latex formula], an annual installment of value [unparseable or potentially dangerous latex formula], an interest accumulation of value [unparseable or potentially dangerous latex formula], and a number of annual installments (and years of interest as well) [unparseable or potentially dangerous latex formula].

We see that the total amount of money obtained after [unparseable or potentially dangerous latex formula] years (and thus installments) is

[unparseable or potentially dangerous latex formula]

for the reason that we have [unparseable or potentially dangerous latex formula] installments of value [unparseable or potentially dangerous latex formula], and an added interest of [unparseable or potentially dangerous latex formula].

The total amount of money obtained is also:

[unparseable or potentially dangerous latex formula]

(if you don't see this right away, start working with the problem given in the study guide, you should stumble across something like 1.06(1.06(1.06(1.06P+P)+P)+P)=4P+500, and then simplify the terms on the left side)

Then, we have the equation

[unparseable or potentially dangerous latex formula]

Solving for [unparseable or potentially dangerous latex formula], we obtain

[unparseable or potentially dangerous latex formula]

which in turn rearranges into

[unparseable or potentially dangerous latex formula]

and finally can be rewritten as your solution:

[unparseable or potentially dangerous latex formula]

So yeah, it's not exactly intuitive...but, it works.