# 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.

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

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

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

$$c=\Big(1,\dfrac{3}{2},2,\dfrac{5}{2},3,\ldots\Big).$$

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

In $$a$$,

$$\begin{array}{c} \underbrace{2, 5} \\\\ 3 \end{array}$$ $$\begin{array}{c} \underbrace{5, 8} \\\\ 3 \end{array}$$ $$\begin{array}{c} \underbrace{8, 11} \\\\ 3 \end{array}$$ $$\begin{array}{c} \underbrace{11, 14} \\\\ 3 \end{array}$$

In $$b$$,

$$\begin{array}{c} \underbrace{3, 1} \\\\ -2 \end{array}$$ $$\begin{array}{c} \underbrace{1, -1} \\\\ -2 \end{array}$$ $$\begin{array}{c} \underbrace{-1, -3} \\\\ -2 \end{array}$$ $$\begin{array}{c} \underbrace{-3, -5} \\\\ -2 \end{array}$$

In $$c$$,

$$\begin{array}{c} \underbrace{1, \dfrac{3}{2}} \\\\ \dfrac{1}{2} \end{array}$$ $$\begin{array}{c} \underbrace{\dfrac{3}{2}, 2} \\\\ \dfrac{1}{2} \end{array}$$ $$\begin{array}{c} \underbrace{2, \dfrac{5}{2}} \\\\ \dfrac{1}{2} \end{array}$$ $$\begin{array}{c} \underbrace{\dfrac{5}{2}, 3} \\\\ \dfrac{1}{2} \end{array}$$

In all three cases we notice that these differences are always the same: $$3$$ in the first succession, $$-2$$ in the second one and $$\dfrac{1}{2}$$ 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 $$(a_n)_{n\in\mathbb{N}}$$, 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.

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