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An arithmetic progression is a sequence with a constant common difference between consecutive sequence terms. It has the form $a,a+r,\dots ,a+(m-1)r$ and is nonconstant when $r\ne 0$. Using $d$ for the common difference and numbering the first sequence term as $a_1$, its explicit and recursive descriptions areThe 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.^[ref]

For an original example, start at seven and add three each time:The eighth entry is $7+7\times 3=28$: 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 $1/2,3/4,1,5/4,\dots$.

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.^[ref]