Thread: Whatever Joey does blog View Single Post
2019-04-05, 08:21   #108
Nick

Dec 2012
The Netherlands

2·751 Posts

Quote:
 Originally Posted by jvang Is this a coincidental correlation or does it have some meaning?
Good observation!

Here's a picture to see what's going on:

If we add this to the observation you made earlier we have 2 sequences:
$\begin{eqnarray*} 1+2+3+\ldots +n & = & \frac{1}{2}n(n+1) \\ \underbrace{1+3+5+\ldots +(2n-1)}_n & = & n^2 \end{eqnarray*}$
In both sequences, the difference between terms next to each other remains constant
(the numbers go up by 1 each time in the first sequence and by 2 each time in the second one).

Sequences with this property are known as arithmetic progressions.
There is a formula for the sum of all the terms in any arithmetic progession, which is useful to know:
add the first and last terms together, multiply by the number of terms and divide by 2