This is a question based on the exponential growth pattern of a game mechanic in World of Warcraft: The Heart of Azeroth item in the game has a level that can be increased by gathering Azerite, with each level requiring roughly 1.3 times as much Azerite as the one before. I'm trying to calculate how much Azerite I have left to gather to get from my current level (40) to the max level (50) when I want to start playing again, so effectively $$X + 1.3X + 1.3^2 X + 1.3^3 X + \ldots + 1.3^{10}X$$
I'm trying to figure out a formula that I can use so I don't have to manually add these numbers (even though deriving that formula would probably be slower than just adding the 10 numbers manually). Essentially, given a start Azerite requirement of X, a rate of increase of p and n numbers to sum, is there a simple formula to calculate this exponential sum without having to calculate and add each number? I assume there is one, but I haven't done any real math since high school so I have no idea where to start.
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$\begingroup$How to get the formula...
let $S$ be the sum of the series.$$S=x+px+p^2x+p^3x+ \dots + p^nx$$Then $$pS = px+p^2x+p^3x+ \dots + p^nx + p^{n+1}x$$$$S-pS = (1-p) S = x - p^{n+1}x$$
Therefore, $$S = \frac{x(1-p^{n+1})}{1-p}$$
$\endgroup$ $\begingroup$You are looking at a geometric series$$ x + px + p^2x + \ldots xp^n = x\sum_{k=0}^n p^k = \frac{x\left(1-p^{n+1}\right)}{1-p} $$
If $|p|<1$, $n$ is large and you are happy with an approximation, $p^{n+1}$ becomes very small, so your sum is approximately $$\frac{x}{1-p}.$$
For your example this does not approximate well since $p = 1.3 > 1$...
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