We begin our journey with triangular numbers, the foundation of figurate numbers.
At stage n, points are arranged on n line segments parallel to each side of the triangle (which may be of any shape), making 3n segments in all. The number of points, *Tn = 1 + 2 +…+ n, can be calculated in various ways, but one of the most elegant is surely the proof without words from Irving Adler’s book Nombres et figures*, reproduced below.
This n by n + 1 rectangle is divided into two congruent triangular arrays of order n, giving Tn=n(n+1)2.T_n = \dfrac{n(n+1)}{2}.
Three construction methods ----------------------------