Editing Triangular number (section)

==Triangular roots and tests for triangular numbers{{Anchor|Triangular root}}==
By analogy with the [[square root]] of {{mvar|x}}, one can define the (positive) triangular root of {{mvar|x}} as the number {{mvar|n}} such that {{math|1=''T<sub>n</sub>'' = ''x''}}:<ref name="EulerRoots">{{Citation |last1=Euler |first1=Leonhard |author-link=Leonhard Euler |last2=Lagrange |first2=Joseph Louis |author2-link=Joseph Louis Lagrange |year=1810 |title=Elements of Algebra |edition=2nd |volume=1 |publisher=J. Johnson and Co. |pages=332–335|title-link=Elements of Algebra }}</ref>
<math display=block>n = \frac{\sqrt{8x+1}-1}{2}</math>

which follows immediately from the [[quadratic formula]]. So an integer {{mvar|x}} is triangular [[if and only if]] {{math|8''x'' + 1}} is a square. Equivalently, if the positive triangular root {{mvar|n}} of {{mvar|x}} is an integer, then {{mvar|x}} is the {{mvar|n}}th triangular number.<ref name="EulerRoots" />