There is wiki that goes into the basics of geometric sums and their basic identies on Wikipedia: http://en.wikipedia.org/wiki/Geometric_series. For now I only present to you one of the places I have run into this little bad boy, and that is in the calculation of amortization. Every time I go through a application that does this I wondered where the math came from, I finally decided to work it out for my self.

Lets start with the knowns,

*r*is the yearly APR,

*t*is the number of times the APR is compounded in a year,

*n*is the number of years,

*P*is the principle on the loan, and

*D*is your down payment. The only variable we are looking for

*X*is your monthly payments. At the end of the loan term or n*t payments we would like there to be no balance in the account.

First the amount that in the loan has to start with:

Each subsequent payment takes interest from the previous period, and deducts your payment:

Expanding the recursive relationship manually reveals a equation that can have Horner's Rule applied to reduce it to a more manageable form:

Transposing by Horner's Rule and collapsing the subsequent series into a Geometric Series:

Apply the identity from the Wikipedia page above:

Recalling that at the end of the term we desire no net balance, solving for X and simplifying:

Now I have the magic amortization formula in my back pocket in case I ever need it again. You can also quickly calculate your total payments

*Xnt*or the amount of interest you paid

*Xnt - (P - D)*. The two other areas I have had the Geometric Series show up was calculating the probability of winning at a game of dice, and a 401k what if Excel sheet I made. The Dice game is really interesting because it deals with an unbounded Geometric Series.

UPDATE: A simplification of the formula above is possible:

If you know how much you can pay each month

*X*and are looking for how much of a mortgage you can afford you can resolve the equation above for

*P-D*

Also note that there is a linear relationship between the amount you can pay and the amount of a mortgage you can afford.

UPDATE: Homer's Rule replaced with Horner's Rule

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