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
Yes, these are indeed the triangular numbers! Triangular numbers can be represented by an equilateral triangle, and each row of the triangle corresponds to the sum of consecutive natural numbers.
To continue the pattern, we can calculate the sum of the sixth row. According to the formula you provided:
nth number = (n+1)(n)/2
For the sixth row, n = 5. Plugging it into the formula:
(5+1)(5)/2 = 6*5/2 = 30/2 = 15
So, the sum of the sixth row is 15.