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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.

1 answer

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.
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