Explain triangular numbers

Answer 1

Just as square numbers represent the number of dots in a square with a certain number of dots on each side, triangular numbers represent the dots that make up different sized triangles.

The sequence that defines these numbers is [1 + 2 + 3 + ... + (n - 1) + n], as there is one dot at the top of the triangle, two dots in the next row, three in the next row, and so on (think of the setup for tenpin Bowling - ten is the fourth triangular number (1 + 2 + 3 + 4 = 10)).

Just as squares have an algebraic representation (x2) as well as a geometric one, triangular numbers can be expressed as (x2 + x)/2 - this can be proven by induction (algebraically), or geometrically. There are other polygonal numbers such as pentagonal and hexagonal numbers. The algebraic representation of these can be found by expressing them as a sum of triangular numbers (based on their geometric representations)

Interestingly, the sum of two consecutive triangular numbers, is always a square number. This can be shown geometrically or algebraically as follows:

(x2 + x)/2 + [(x + 1)2 + (x + 1)]/2 = [x2 + 2x + 1 + (x + 1)2]/2

= 2(x + 1)2/2

= (x + 1)2

So ALL polygonal numbers are dependent on triangular numbers!

Hope this helps,

Nick :)