= Triangular number
{title2=$T_n=n(n+1)/2$}
{wiki}
A <triangular number> counts rows of $1,2,\ldots,n$ objects, with $n$ a <nonnegative integer>:
$$
T_n=\sum_{k=1}^{n}k=\frac{n(n+1)}2,\qquad T_0=0.
$$
The first values are $0,1,3,6,10,15,21,28,36,45,\ldots$. 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 $n\geq1$. The row lengths form an <arithmetic progression>, while their running totals are <triangular numbers>.
\Image[/triangular-numbers-dot-pairing.png]
{title=Two copies of six triangular rows form a rectangle of six rows and seven dots}
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, \b[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 $T_4=10$.
\Image[/triangular-numbers-pairs-and-square.png]
{title=Ten unordered pairs of five objects, and two consecutive triangular counts filling a square}
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>.<Introduction to Arithmetic by Nicomachus (Hahn teacher edition)>[Book II, Chapters 8, 12 and 20, printed pp. 63–64, 70 and 75]{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.
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