Question

There's a question I've been having difficulty solving, and I would really appreciate if you can show he steps to how I can achieve the answer.

QUESTION: The numbers 1, 4, 10, 20, and 35 are called tetrahedral numbers because they are related to a four sided shape called a tetrahedron. Determine a mathematic model that you can use to generate the nth tetrahedral number.

Answer 1

The triangular numbers are
1,3,6,10,...

So, the nth tetrahedral number is the sum of the first n triangular numbers.

The nth triangular number is the sum of the first n integers (1+2+3+...) = n(n+1)/2

It is clear that the nth tetrahedral number will be a cubic expression in n.

The nth integer is n/1!
The nth triangular number is n(n+1)/2!
The nth tetrahedral number is n(n+1)(n+2)/3!
and so on.

This can easily be proven using induction.