Ideas in Geometry/Instructive examples/Section 2.4 number 23
For this problem you need to look at the picture in relation to n^2/2 + n/2 = n(n+1)/2. This picture shows a triangle with equal squares along the base and the height. I will label this number n. If we square n we get n^2. This would be the area of a square. But since we only wants n^2/2, and we have a half of a square, we can divide n^2/2. This gives us the first part of the equation we are looking for. The second part comes from the 6 half triangles. There are 6 dark grey figures, and 6 is our n value. But since we only have 6 halves we need to divide n/2. This gives us our second part of the equation. When we put the two parts we found together we get n^2/2 +n/2. Because these fractions have common denominators we can add them to get n^2+ n/2. We can factor the top part (n^2+n) to get n(n+1) and find that it equals n(n+1)/2. This is what we are trying to prove.