what are triangular number's

Answer 1

Triangular numbers are the numbers you get when you arange items in a triangular shape, each row having one more than the row above it.

Top row: 1
Next row: 2
...
So, the nth triangular number is 1+2+3+...+n = n(n+1)/2

Similarly, square numbers are what you get when you arrange items in an nxn square. If you have a square of side n, then the next square has n items above and to the side, plus 1 at the new upper corner.

So, Sn+1 = Sn + 2n+1

You get them by adding up odd numbers:

1
1+3 = 4
1+3+5 = 9
1+3+5+...+(2n-1) = n^2