Arithmetical progression: definition

An arithmetical progression is a kind of succession, i.e., a sorted and infinite collection of real numbers, in which every term is obtained adding a constant quantity to the previous one.

Example

If we consider the successions that have the following first terms:

$a = (2, 5, 8, 11, 14, \dots),$

$b = (3, 1, - 1, - 3, - 5, - 7, \dots),$

$c = (1, \frac{3}{2}, 2, \frac{5}{2}, 3, \dots) .$

And, in each of them we calculate the difference between every term and the previous:

In $a$ ,

$\begin{matrix} \underset{⏟}{2, 5} \\ 3 \end{matrix}$ $\begin{matrix} \underset{⏟}{5, 8} \\ 3 \end{matrix}$ $\begin{matrix} \underset{⏟}{8, 11} \\ 3 \end{matrix}$ $\begin{matrix} \underset{⏟}{11, 14} \\ 3 \end{matrix}$

In $b$ ,

$\begin{matrix} \underset{⏟}{3, 1} \\ - 2 \end{matrix}$ $\begin{matrix} \underset{⏟}{1, - 1} \\ - 2 \end{matrix}$ $\begin{matrix} \underset{⏟}{- 1, - 3} \\ - 2 \end{matrix}$ $\begin{matrix} \underset{⏟}{- 3, - 5} \\ - 2 \end{matrix}$

In $c$ ,

$\begin{matrix} \underset{⏟}{1, \frac{3}{2}} \\ \frac{1}{2} \end{matrix}$ $\begin{matrix} \underset{⏟}{\frac{3}{2}, 2} \\ \frac{1}{2} \end{matrix}$ $\begin{matrix} \underset{⏟}{2, \frac{5}{2}} \\ \frac{1}{2} \end{matrix}$ $\begin{matrix} \underset{⏟}{\frac{5}{2}, 3} \\ \frac{1}{2} \end{matrix}$

In all three cases we notice that these differences are always the same: $3$ in the first succession, $- 2$ in the second one and $\frac{1}{2}$ in the third one.

In other words, every term is obtained by adding the same number to the previous one .

Giving a formal definition, we will say that an arithmetical progression $(a_{n})_{n \in N}$ , is a succession in which the difference of every term with the previous one is constant, that is:

$a_{n + 1} - a_{n} = d$

when $a$ is any natural $n$ . We will call the constant $d$ as the difference of the progression.

Example

The differences of the progressions $a$ , $b$ and $c$ are, respectively, $3, - 2$ and $\frac{1}{2}$