Friday, May 23, 2014

Al Kharaji Summation

Given the summation
find an equation that defines it.
I start by constructing the sequence xn.
 That is,
Using exploration and the series, I derived an equation that defined the above summation:

 See scratch work below:

First I tried different values of n. That resulted in the circled numbers: 1, 9, 36, 100, and 225. I noticed that all these numbers were perfect squares with the corresponding numbers 1, 3, 6, 10, 15, 21 as their square roots. These numbers form a sequence defined as xn above. I used this sequence to derive a way to find the summation of any n.

1 comment:

  1. I think this is a good topic for your post, but I can't see the scratchwork image (asked me for a gmail address!) where most of the thinking is. (complete)
    consolidation: be good to tie this up with your personal perspective. One framework we use is: what? (recap of important bits; not necessary here I don't think), so what? (why is this important?), now what? (given this work, what's next).
    clear, coherent, content: + (as far as I can tell.)