triangular numbers

The triangular numbersMathworldPlanetmath are defined by the series


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


The name triangular number comes from the fact that the summation defining tn can be visualized as the number of dots in

where the number of rows is equal to n.

The closed-form for the triangular numbers is


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


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 nth triangular number is less than or equal to the nth 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 nn or n2), and then use the first few triangular numbers to solve the general 2nd degree polynomial An2+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 formulaMathworldPlanetmathPlanetmath 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