Relation involving triangular number and othersequences

A triangular number may be written as series of natural numbers. Its particular behavior is observed if it be represented as dots arranged to form a triangle (as follows in the illustration below). However, series of natural numbers might be considered sums of arithmetic progressions and hence may be reduced to a simple relation involving the number of elements and their first and last elements. That is, in mathematical language:
eq1
 1

And restrictively for triangular numbers (natural numbers series):

eq2

Some complicated series and sequences are easily imagined through triangular numbers, by their simplicity. Here will be shown three examples of how triangular numbers can represent them.

The first example involves sequences of graphs whose have a form of a regular polygon with all their vertex connected. Notice that, from an nth graph from the sequence, the number of connections are equals to a triangular number (as in the image below) with an inferior index. Thus:
eq2.1
2

Wherein Cn is the number of connections and Tn−1 means a triangular number with inferior index relative to Cn. However, as any triangular number may be written as an arithmetic progression follows that:

eq3

Triangular numbers can also represent cubic series, which are, obviously, in the form:
eqbc.
An interesting property of this series is that it is always a square. Therefore if this series would be imagined as a correspondent number of dots, the number of dots would fit in a square (as illustrated below).  Analyzing those squares, is founded a surprising fact that their sides are exactly triangular numbers. And hence is possible to infer that:

eq4

3

Which also might be written as:

eq5

Certainly, if triangular numbers represent cubic series, it can represent quadratic series as well:
eq5.5.
Here will be shown a proof developed by Jim Fowler in his online course “Calculus One” presented in the site coursera.org. First, all the squares (being considerate dots again) are organized one above other forming a pyramid. After the anterior process is repeated inversely and next to the first pyramid, forming a rectangle with width equals to the nth term of the series plus one and height equals to an nth triangular number. Therefore the area of this square is:

eq6

4

 

Wherein, if divided by 3, is correspondent to the series, hence:

eq7

Or:

eq8

Determining a triangular number as a arithmetic progression

A triangular number  may be organized as a rectangle of dots instead of a triangle. Notice that each row of this rectangle is always equals to (n+1) and there are n/2 lines. Hence a triangular number can be written as the area of this rectangle, which is:

eq9

5

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