A triangular number or triangle number is the number of dots in an equilateral triangle uniformly filled with dots. For example, three dots can be arranged in a triangle; thus three is a triangle number. The nth triangle number is the number of dots in a triangle with n dots on a side.
the nth triangular number is given by n(n+1)/2.
the sum of two consecutive triangular numbers is a square number with the sum being the square of the difference between the two.
Tn + T(n+1) = n2
Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n..
cause [n(n+1)/2]2 = sum of all the cubes of the individual natural numbers from one 1 to n.
if a perfect number is a whole square then, n(n+1)/2 = m2
for example 1 and 36.
also all perfect numbers are triangular.