triangular numbers

The triangular numbers are defined by the series

t_{n}=\sum_{i=1}^{n}i

That is, the $n$ th triangular number is simply the sum of the first $n$ natural numbers. The first few triangular numbers are

1,3,6,10,15,21,28,\ldots

The name triangular number comes from the fact that the summation defining $t_{n}$ can be visualized as the number of dots in

\begin{matrix}\bullet&&&&&&&\\ \bullet&\bullet&&&&&&\\ \bullet&\bullet&\bullet&&&&&\\ \bullet&\bullet&\bullet&\bullet&&&&\\ \bullet&\bullet&\bullet&\bullet&\bullet&&&\\ \bullet&\bullet&\bullet&\bullet&\bullet&\bullet&&\\ \vdots&&&&&\vdots&\ddots&\end{matrix}

where the number of rows is equal to $n$ .

The closed-form for the triangular numbers is

t(n)=\frac{n(n+1)}{2}

Legend has it that a grammar-school-aged Gauss was told by his teacher to sum up all the numbers from 1 to 100. He reasoned that each number $i$ could be paired up with $101-i$ , to form a sum of $101$ , and if this was done $100$ times, it would result in twice the actual sum (since each number would get used twice due to the pairing). Hence, the sum would be

1+2+3+\cdots+100=\frac{100(101)}{2}

The same line of reasoning works to give us the closed form for any $n$ .

Another way to derive the closed form is to assume that the $n$ th triangular number is less than or equal to the $n$ th square (that is, each row is less than or equal to $n$ , so the sum of all rows must be less than or equal to $n\cdot n$ or $n^{2}$ ), and then use the first few triangular numbers to solve the general 2nd degree polynomial $An^{2}+Bn+C$ for $A$ , $B$ , and $C$ . This leads to $A=1/2$ , $B=1/2$ , and $C=0$ , which is the same as the above formula for $t(n)$ .

Title	triangular numbers
Canonical name	TriangularNumbers
Date of creation	2013-03-22 12:16:08
Last modified on	2013-03-22 12:16:08
Owner	akrowne (2)
Last modified by	akrowne (2)
Numerical id	6
Author	akrowne (2)
Entry type	Definition
Classification	msc 11A99
Classification	msc 40-00