6. The triangular numbers are obtained by adding up consecutive whole numbers starting with one. List the first 10 triangular numbers.
1 = 1
3 = 1 + 2
6 = 1 + 2 + 3
10 = 1 + 2 + 3 + 4
15 = 1 + 2 + 3 + 4 + 5
21 = 1 + 2 + 3 + 4 + 5 + 6
28 = 1 + 2 + 3 + 4 + 5 + 6 + 7
36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8
45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 6 + 9 + 10
The reason that these are called the triangular numbers is because they tell us how many objects we would have if the objects were arranged in a triangle, if we knew how many objects were on the side of the triangle
Notice that these numbers are found in Pascal's triangle.
This is because they are obtained by adding up the consecutive numbers in the tier above.
To see how these triangular numbers relate to the number of two element subsets of a set, see the solution to problem 17.