On arithmetic progression

Introduction

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: \[\label{eqn:vala} a_n=\frac{n(n+1)}{2}+1\]