Can you show me the tetrahedral number formula and explain it simply?

Question

I was given the formula and told to find the position of the tetrahedral number 1771. I know it is the 21st number by writing them all out which took a long time. Everywhere explains the formula really difficultly can you work through it for me please.

Answer 1

The n(th) tetrahedral number is given by :

n(n + 1)(n + 2) / 6

so we need to solve : n(n + 1)(n + 2) / 6 = 1771

or, n(n + 1)(n + 2) = 6 * 1771.

It's not too hard to find the factors of 1771, so we have :

n(n + 1)(n + 2) = 6 * 7 * 11 * 23

Now factor the 6 a bit further :

n(n + 1)(n + 2) = 2 * 3 * 7 * 11 * 23

Now, can we combine the numbers in a way

that will produce 3 consecutive numbers?

Yes, 3 * 7 = 21 and 2 * 11 = 22 and we already have 23.

So, n(n + 1)(n + 2) = 21 * 22 * 23

Therefore, n = 21.

This is much easier than solving a

cubic equation by the usual methods.