UC Berkeley StarCraft Class, Week 4

I was not able to attend the week 3 class, but here's the summary of week 4.
Professor Feng started with an analogy of the game Battleship (you know, that game where you say "H2," then the other guy says "You sunk my battleship!") What if you played a game of Battleship where the number of attacks you get per turn is equal to the number of ships you have left, he asked. Feng is pointing out the essential slippery slope nature of the game, that your ability to attack is reduced as you start fall behind (as opposed to many games where your ability to attack is unaffected by falling behind--I wrote about this here).
Perfect Micro
Then we explored the math behind this idea. To make it easier, we considered the damage done between two packs of marines assuming all marines are within range of each other in an idealized situation. One player has N1 marines versus the other player's N2 marines. Assume the first player has perfect micromanagement while the other player has the worst micromanagement possible. In perfect micromanagement, as many of your units as possible deal damage for as long as possible. In other words, you focus fire on a single enemy marine and kill him as soon as possible so that the enemy's damage output is reduced. You then immediately switch to a new marine, focus fire on him to kill, switch targets, and so on. Meanwhile, your opponent is attacking in the worst way possible: he spreads out his damage evenly amongst your marines, not killing any of them (thus allowing you to keep your overall damage per second high while his declines).
If each marine deals D damage per shot, then after volley 1, player 1 dealt N1D damage while player 2 dealt N2D damage. Player 1 killed N1D/K marines where K = the hit points of a marine. Player 2 killed 0 marines though.
After a second volley, player 1 still deals N1D damage and again kills N1D/K marines. Player 2 only has N2 - (N1D/K) marines left though, so he deals (N2 - (N1D/K)D damage and kills 0 marines again.
After m volleys of this, how many marines are left on each side? Player 1 will have the same number of marines he started with (N1) for a long time, then they will all suddenly die at about the same time. This is because the opponent is attacking in the least efficient way possible here, basically keeping player 1's marines alive as long as possible. Calculating player 2's remaining marines is more tricky though. Player 2 will deal this much damage after m volleys: