Trapezoidal numbers form an important class of figurate numbers that has extensively been studied (see [2-7]).
In Section 2, general properties of m-trapezoidal numbers are discussed as well as links with other figurate numbers. In Section 3, the characterization and enumeration of 2-trapezoidal numbers have been given together with some illustrative examples.
In this section, we focus on general properties of m-trapezoidal numbers and links with other figurate numbers such as triangular numbers, trapezoidal number, and rectangular numbers.
Triangular numbers are
figurate numbers because they represent counting numbers as a geometric configuration of equally spaced points.
"The Mystery Of Numbers" is both a numerical and spiritual journey: starting from prime and
figurate numbers; to Fibonacci sequence and the golden section; to alchemy and the Mayan calendar; to the atoms and its forces, along with the ether and the fourth dimension.
Playing with
figurate numbers can help making triangle and other geometric number-shapes.
Teachers might ask, for example, "What sequence of moves allows the first (or second) player to make the earliest possible harvest?" Even without such investigations, early experience playing oware can provide both visual and tactile foundations for more formal explorations of certain mathematical ideas, including
figurate numbers and their properties, iterative and self-referential processes, and cellular automata.
This is exemplified in the student notes booklet of the "Euler Enrichment Stage" in a section that deals with
figurate numbers. The dot patterns of counting numbers, triangular numbers, square numbers and pentagonal numbers connect into ideas associated with sequences and algebra.
We pick up from there and use another neat way of adding lists of numbers to find a way of getting general formulae for figurate numbers and use Gauss's method to check it!
If in the previous issue of AMT (page 15) you didn't try to test Fermat's declaration about figurate numbers being used to express any number, then you might be in a better position to do so now, having found a few more types of figurate numbers with which to play.
Its definition is linked in history to our investigation of figurate numbers. Here is a bit of information on this special word and a neat exercise involving our earlier set of numbers, the odd numbers, that we will apply later to these triangular numbers and see what happens.
Figurate numbers were a concern of Pythagorean geometry, (I just love these links between numbers and shapes!) since Pythagoras is credited with initiating them, and the notion that these numbers are generated from a gnomon or basic unit.