Rate of Growth of pyramid structure.

Question

This might be simpler than I'm making it out to be, but how can the growth rate of the pyramid structure be represented.

      1
     / \
    2   3
   / \ / \
  4   5   6
 / \ / \ / \
7   8   9  10

• Firstly I thought its n+1 but that is only compared to the previous row, not the whole dataset.
• Then I though about binary trees (as each node branches into two others, but as they are interconected that doesn't work either).

Any help would be appreciated.

Answer 1

You are looking for the triangular numbers, given by

$$T_n = \sum_{k=1}^n k = 1+2+3+ \dotsb +n = \frac{n(n+1)}{2} = {n+1 \choose 2}$$