Since 2k-1 is the average, shouldn't I use this list: 2k-3, 2k-2, 2k-1, 2k, 2k+1?
That's a 100% perfectly valid way to solve this problem. :-)
But it's good to take note that if you use any 5 consecutive integers, the difference between the least and the greatest is always going to be 4, because the largest is 4 bigger than the first. The (2k-1) in this problem is just a distraction, and we want to learn to recognize when that's the case so we don't waste precious time!