| it is based on the formula (n^2 + n)/2 picture it like finding the area of a trinagle that is half of a square (long side goes from diagonally opposite corners) formula for area of a square is n^2, so we need half that. What about +n part of it? It is because we are dealing in discrete chunks, not smoothly continuous. --- Alternately, imagine a series of children's blocks 1 2 3 now we have a 'triangle' with left side 3 high and bottom 3 wide. IF we make an identical triangle and rotate it round so that they fit together to make a rectangle, it looks like this 1 + 3 2 + 2 3 + 1 This makes a rectangle 3 x 4, and we should be able to figure out that this gives us an area of 12. So the original triangle ( 1,2,3 ) gives us half that and thus the sum from 1 to 3 is 6. So the pattern for summing the numbers 1 through N is to take a square of length N, add 1 to one of its sides, so it has sides of length N and N+1, and then find the area (which is always N^2 + N) and then halve that. |