Ok i figured out the first part that i need to square the numbers so after 4 it will be 5x5,6x6,7x7,8x8and 20x20 to make up the second column, what i don't get is the 3rd one, how do i come up with the number of white triangular shapes used?

Answer 1

I tried to explain it by using the arrangement of billiard balls, but I guess you don't play pool.

let me try to "draw" the billiard balls

. . .o
. . o o
. .o o o
. o o o o
o o o o o o

sum of one row = 1
sum of two rows = 3
sum of three rows = 6
sum of four rows = 10
sum of five rows = 15

difference between first and second = 2
difference between third and second = 3
difference between fourth and third = 4
difference between fifth and fourth = 5

ahhh!

Can you now continue the pattern?
They are called the "triangular" numbers since they form an equilateral triangle.

here is another pattern for your third column numbers

1 = (2x1)/2
3 = (3x2)/2
6 = (4x3)/2
10 =(5x4)/2
.
.
nth number = (n+1)(n)/2