# Triangular Number **A number associated with the amount of objects arrangeable in an equilateral triangle.** ![[TriangularNumber.svg|500]] *The first four triangular numbers without the zeroth* A **triangular number** $T_{n}$ is an example of a figurate number and is the number of dots that can be arranged into an equilateral triangle with $n$ dots comprising a side. The number of edges in a [[complete graph]] $K_{n}$ of $n$ edges is equal to triangular number $T_{n}$. The third diagonal in both directions of [[Pascal's Triangle#Figurate numbers of simplexes|Pascal's triangle]] contains all triangular numbers. ## Formula The formula for the $n$th triangular number is $T_{n}=\sum_{k=1}^{n}k=1+2+\dots+n=\frac{n(n+1)}{2}=\binom{n+1}{2}$ It is equivalent to the sum of the first $n$ natural numbers and the [[Binomial Theorem#Binomial coefficient|binomial coefficient]] $\binom{n+1}{2}$.