Friday, October 10, 2008

Series Part 2

Continuing from the previous post on this subject, lets find the formulas for the sum of the two series I mentioned in the previous post, the arithmetic and the geometric progressions:

Arithmetic progression:
Firstly, lets look on a very simple case - the progression 1+2+3+...+n=S. If we just look on it, it is not very clear what the sum is. However, if we will write it in a slightly different manner the answer will be obvious:


Obviously the sum of all the numbers is 2S. We also know that there are n columns and the sum of any column is n+1. Therefore 2S=n(n+1). From this we get the formula by division.

Now lets look on the general case. The general arithmetic series is:


By the way, this formula requires to know the first number in the progression. However, if you know the last one instead of the first you can also use this formula - after changing d to (-d).

Geometric progression:
Firstly, lets again consider a specific case. Lets look on the sum of the following progression:


Getting from this to the general case is extremely easy. The only thing we need to do is to remember that a general geometric progression is just a multiplication of the case we took care of by a constant number a. Therefore, the formula for the sum is:


In the next post I will talk about infinite series and their sums.


smilitude said...

It was really nice to see the proof of the geometric progression! ":) its so cute!

I am also a math major, first year! :) its nice to see your blog! :)

Anatoly said...

Hello smilitude,
Thanks for stoping by, glad you liked my post.