= Arithmetic progression
{title2=$a_k=a_1+(k-1)d$}
{wiki}
= Arithmetic sequence
{synonym}
An <arithmetic progression> is a <sequence> with a constant <common difference> between consecutive <sequence terms>. It has the form $a,a+r,\ldots,a+(m-1)r$ and is nonconstant when $r\ne0$. Using $d$ for the <common difference> and numbering the first <sequence term> as $a_1$, its explicit and recursive descriptions are
$$
a_k=a_1+(k-1)d,\qquad a_{k+1}=a_k+d.
$$
The explicit formula reaches any position directly; the recurrence reaches the next position from the current one. The factor $k-1$ counts the steps after the first entry. For real $a_1,d$, a positive <common difference> makes the <sequence> increase, a negative one makes it decrease, and zero makes it constant.<Precalculus 2e by Jay Abramson/11.2 Arithmetic Sequences>[Precalculus 2e, §11.2 Arithmetic Sequences: definition, Recursive Formula for an Arithmetic Sequence and Explicit Formula for an Arithmetic Sequence]{ref}
For an original example, start at seven and add three each time:
$$
7,10,13,16,19,22,25,28,\ldots,\qquad a_k=7+3(k-1).
$$
The eighth entry is $7+7\times3=28$: eight <sequence terms> require seven increments. This counting principle makes <arithmetic progressions> useful for regularly spaced positions, repeated equal additions, and iteration over evenly spaced <integers>. The values need not be <integers>; starting at $1/2$ with <common difference> $1/4$ gives $1/2,3/4,1,5/4,\ldots$.
A <geometric progression> instead holds a <ratio> constant. In the displayed <arithmetic progression>, $10/7$ and $13/10$ differ even though $10-7=13-10=3$. Multiplying each entry by two would give a different kind of repeated update from adding three. Distinguishing the two rules matters before predicting later entries or adding the <sequence terms>.
Equal differences also appear in <Nicomachus>'s treatment of arithmetic proportion. He describes successions of at least three numbers whose neighboring differences agree, then explains how regularly omitting counting numbers changes the common gap. This connects the progression to the <arithmetic mean>: a middle entry is equally far from its two neighbors.<Introduction to Arithmetic by Nicomachus (Hahn teacher edition)>[Book II, Chapter 23, §§1 and 3, printed p. 80, PDF page 85; translated mathematical passages, excluding the editor's introductory paragraph and footnotes]{ref}
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