EDIT- Thanks to Jair taylor I got the answer but now my question is- What will be the sum in $p^{th}$ dimension??
Polygonal numbers is a sequence of numbers. The $k^{th}$ term of a n-polygonal series is the minimum number of dots it would require us to make a n-sided polygon with k dots in each side. We all have heard about triangular series.It is one such example known as 3-gon series.
Here is an illustration of how to arrange these dots in a triangular series https://en.wikipedia.org/wiki/Triangular_number in the top right corner you could see the diagram.
The series of square numbers is- 1,4,9,16,25,... This is a 4-gonal series.
Again illustratio for 5-gon series-https://en.wikipedia.org/wiki/Pentagonal_number
You could find more such by searching them. (Octagonal numbers also know as star numbers are interesting).
My question is- Can we derive a formula for the $k^{th}$ term of a n-gon series in general? Can we derive a formula for the sum of first k terms of a n-gon series in general?
These sequences might help to analyze a pattern.
3-gon - 1,3,6,10,15,21,28,36,...
4-gon - 1,4,9,16,25,36,49,64,...
5-gon - 1,5,12,22,35,51,70,92,...
8-gon - 1,8,21,40,65,96,133,176,...