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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]

Translated Introduction to Arithmetic by Nicomachus (Hahn teacher edition); the edition credits the underlying translation to 1926.

Edited Introduction to Arithmetic by Nicomachus (Hahn teacher edition).

For every nonnegative integer $n$,For example, $1^3+2^3+3^3+4^3=1+8+27+64=100=T_4^2$. A short proof uses the triangular number recurrence and the triangular number square-complement identity:Summing this difference of squares identity gives a telescoping sum; all intermediate squared triangular numbers cancel. Nicomachus describes a related grouping of the odd integers: take one, then the next two, then the next three, producing $1,3+5,7+9+11,\dots$, which are $1^3,2^3,3^3,\dots$. This historical grouping is recorded in Introduction to Arithmetic by Nicomachus (Hahn teacher edition) (Book II, Chapter 20, §5, printed p. 75).

A triangular number counts rows of $1,2,\dots ,n$ objects, with $n$ a nonnegative integer:The first values are $0,1,3,6,10,15,21,28,36,45,\dots$. The empty arrangement contributes zero, by the empty sum convention. Adding one more row gives the triangular number recurrence $T_n-T_{n-1}=n$ for $n\ge 1$. The row lengths form an arithmetic progression, while their running totals are triangular numbers.

Pair the rows with a reversed copy: $1+n=2+(n-1)=\cdots =n+1$. There are $n$ such pairs, so two copies contain $n(n+1)$ dots. Dividing by two proves the formula. This rectangle picture explains the factor one-half without requiring a formula for a long summation. For example, one hundred rows contain $T_{100}=5050$ dots.

The same count appears whenever every pair of different objects must be considered once. Five participants make ten possible handshakes; five sensors have ten different pairwise comparisons; a complete graph on five vertices has ten edges. These are the triangular number pair-counting identity, with one fewer row than objects. The distinction prevents a common error: $T_5=15$ counts all pairs from six objects, whereas five objects give $T_4=10$.

The triangular number square-complement identity says that two neighboring triangular numbers fill a square. Combining that identity with the one-row difference makes the difference of squares $T_n^2-T_{n-1}^2$ equal to $n^3$. It leads to Nicomachus's theorem: the sum of the first $n$ cubes of numbers is the square of a number $T_n^2$. A change in the way one groups objects exposes a relation between summation, multiplication and geometry.

This geometric view is already visible in Nicomachus's Introduction to Arithmetic by Nicomachus (Hahn teacher edition): Book II, Chapter 8 describes growing triangular rows, Chapter 12 combines successive triangular counts into squares, and Chapter 20 groups successive odd integers into cubes of numbers.^[ref] The linked edition uses Martin Luther D'Ooge's translation, revised with modern teaching material by Nathanael Hahn. Our empty-row index $T_0=0$ is a modern convention; the historical passage starts from unity and calls three the first actual triangle.

For $n\ge 1$,An $n$-by-$n$ array of cells has $n^2$ cells. One side of the diagonal, including the diagonal itself, contains $1+2+\cdots +n=T_n$ cells; the other contains $1+2+\cdots +(n-1)=T_{n-1}$. For $n=5$, the two counts are fifteen and ten, giving twenty-five. The identity is stated in Introduction to Arithmetic by Nicomachus (Hahn teacher edition) (Book II, Chapter 12, §§1–2, printed p. 70).