Today in class we were analyzing the number of half-spaces created by $n$ number of planes. For two planes there are 4 spaces, 3 there are 8, 4 there are 15, etc. our teacher challenged us to find the formula for $n$ planes. Me and my friend came up with $ 1+\sum^n_{x=1} \left(\frac{x(x+1)}{2}+1\right) $ Because based on that when a line cuts a plane in half, the formula for the number of half-planes it creates is $ \frac{n(n+1)}{2} + 1 $ and the difference in the number of half-spaces between each $n$ plane is equal to adding on the same number of half-planes.
+--#of planes--+--1--+--2--+--3--+--4--+--5--+--50--+ |separates into| 2 | 4 | 8 | 15 | 26 |20,876| | _ half-spaces| | | | | | | +--difference--+-----2-----4-----7----16----etc-----+
My teacher said that this was correct, but it would be better if it was a formula/function, where you plug in the variables rather than have to evaluate the summation. I know that the final answer is $ \frac{n^3+5n+6}{6} $ but I need to show my work, and am unsure of how to get from a sum to that formula. Am I approaching this incorrectly? How was the original formula derived?