The {n}th tetrahedral number is the sum of the first n Triangular numbers. Triangular numbers are the sum of the first n integers (1, 3, 6, 10, 15, etc.), and taking the sum of the triangular numbers results in the tetrahedral numbers (1, 4, 10, 20, 35, etc.)
The fifth tetrahedral number is the sum of the first five triangular numbers. The first five triangular numbers are 1, 3=1+2, 6=1+2+3, 10=1+2+3+4, and 15=1+2+3+4+5, so the sum of these numbers is 1+3+6+10+15= 35.
by EmeraldofSeattle March 2, 2007