On arithmetic progression


define the set A

This section defines the set A. the following sections use this definition to study the set itself and study the consequences on other similar sets. But first let us define the set:

\[A=\{a_n=a_{n-1}+n;a_0=1;n\in \mathbb{N}\}\]

This set is infinite and starts at 1. The following table gives an overview of the first elements:

First elements of set A
\(a_n\) \(n\)
1 0
2 1
4 2
7 3
11 4

If we look at these numbers we see a clear resemblance with the triangular numbers. By comparing both we see clearly that every number is equal to the triangular number plus one (more on this in section 4). So we can calculate every element \(a_n\) with the formula: \[\label{eqn:vala} a_n=\frac{n(n+1)}{2}+1\]

This concludes this section where we defined the set A. In the following sections we will learn more on the properties of this set and its relation to other sets.

Prove set A has no arithmetic progression

Now we have defined the set A, this section proves that the set A has no arithmetic progression. To prove this we first have a look at a new set B. We determine when this set B contains elements with arithmethic progression, and then we compare this set with set A. Based on this comparison we determine if A has any form of arithmetic progression.