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.
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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.
n is not an index... it's the last number. I wasn't explicit, sorry. :)