If you look over in the right sidebar, under Memes For You, you will find the expression If not now, when? Here is how I apply this expression to mathematics: During the course of work, if I encounter a mathematical expression, algorithm, or symbol which I don’t fully understand, I take some time and learn it right now. Probably I will run into this mathematical truth again. Perhaps I should have learned this mathematical truth in High School, or College. Plausibly I am missing out on a beautiful gem of mathematics. A few examples: the triangle inequality (simple nearly to the point of triviality, yet beautiful, and vital in certain proofs), the quadratic equation (immensely practical, yet also historical (you can’t use a formula to solve equations of any higher degree), why did I not memorize it in high school?), the definition of real numbers (my favorite definition, centuries in the making).
That leaves us right now with Finite Geometric Progressions. Somehow, they avoided me, or I avoided them, for all of these years. But, here they are, at the root of financial mathematics. I may be able to learn financial mathematics without them, but why should I miss out on an opportunity to become friends with these cute little critters? Here we go.
Starting a month from today, you are going to deposit one dollar each month into an account that pays 0.25 % interest per month. How much money will be in the account in 6 months, at the time of the last payment?
Let’s work backwards. You have made 6 payments.
- The 6th payment has accumulated no interest, so is still worth 1.
- The 5th payment has accumulated one period of interest, so is worth
- The 4th payment has accumulated two months of interest, so it is worth
- The 3rd payment has accumulated three months of interest, so it is worth
- The 2nd payment has accumulated four months of interest, so it is worth
- The 1st payment has accumulated five months of interest, so it is worth
The sum of all the deposits, plus the interest, is hence
We might have written this as or the sum of the first n terms of a geometric progression with common ratio r. In high school, we should have learned that this sum is equal to:
In our case, r = 1.0025, n = 6.
We might also write this as
Which clearly equals
Which mysteriously is also s angle n, or the accumulated value of the annuity immediate.