This should be an easy one for math geeks, but I’m severely math-impaired and I can’t figure it out.
You are given 600 tasks. After completing all 600, you achieve Rank #8 - Master. You receive Rank #1 - Novice, just for showing up for your first task. However, the number of tasks you must complete between ranks is not uniform; rather, it grows by y% between ranks. So, for example, to achieve Rank #2 - Initiate, you must complete x tasks; to achieve Rank #3 - Tenderfoot, you must complete x + y%; to achieve Rank #4, you must do y% more than you did to achieve Rank #3; and so on. For the sake of shorthand y will be called the “difficulty factor.”
Here’s an example, with a starting point of 40 (to achieve Rank 2) and a “difficulty factor” of 40%:
…so you can see I picked the wrong starting point AND the wrong “difficulty factor,” since Rank 8 != 600.
In order to make Rank 8 equal 600 (±1), what should be my starting point for Rank #2 (Rank #1 will always equal 0), and what should be my difficulty factor?
ETA: Don’t consider the difference between Rank #1 (0) and Rank #2 (the starting point) as a factor in this problem.
You can’t solve this problem with the information given. There are two unknowns, x and y, while there is only one equation. You’ll have to specify either one or the other.
You have six ranks to gain to go from 2 to 8. You want to multiply 30 by x 6 times to reach 600. So your difficulty factor should be the 6th root of (600/30), that is, 1.65 more or less.
In general your difficulty factor must be the 6th root of (300/y) where y is your second level threshold. You can work this out easily with a calculator or MS Excel.
If you want your starting rank to be 40, your “difficulty factor” is 57.04%
If you want your starting rank to be 30, your “difficulty factor” is 64.75%
The formula is (A)x[sup]6[/sup] = 600, where A is your starting factor and 6 is the number of steps between your starting number and 600. Solve for x.
There’s a whole family of solutions. In general, you’ll have rank n = rank 2 * (1+x)^(n-2). So if, for instance, you want rank 8 to be 600, and rank 2 to be 30, then 600 = 30*(1+x)^6, or 20 = (1+x)^6, or (1+x) = 20^(1/6), 1+x = 1.6475, giving a final answer of x = 64.75%. So your rank structure would be
1: 0
2: 30
3: 49
4: 81
5: 134
6: 221
7: 364
8: 600
HeyHomie, are you sure you’re asking the right question? You realize this means that to go from rank one to two, it takes 30 tasks, but only 19 to go from two to three. Unless it takes 49 more tasks to get to level 3. If you want a total of 600 tasks to get to level 8, then that’s a separate question. You wouldn’t want a 0-30-49 method for the above reason.
ETA: On review of the OP, I see you want to have 600 done by level 8, not have to do 600 to get from 7 to 8. That means you’d do 30 for the first promotion and 19 for the second. That makes no sense, unless you want it to be that way.