Problem: The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence. Solution: This problem is tricky, but leads a useful lesson in not starting too quickly. Here was my approach. I noted that $22^2 < 500 < 23^3$ and that $7^3 < 500 < 8^3$. I noted that $1^2=1^3$ and $8^2 = 4^3$ (these are the only overlaps) and so the answer is $500+22+7-2$ because for each square or cube, we skip to the next highest integer. This answer is wrong and is off by two. The tricky part about this problem is that this ignores the fact that by our sequence increasing, we are adding new candidates that could be squares or cubes we are not accounting for, so we need a new approach. Specifically, we only know that $x - y = 500$..