How do I notate this?

Question

Let’s say that I have a whole pie, and I take $75\%$ (or 3/4ths) of that pie. Then, I take $75\%$ of the remaining quarter of the pie and add it to the original $75\%$, I would have $93.7\%$ of the pie. I’ll keep on taking turns taking $75\%$ of the smaller portion and adding it to the larger portion. How would I write a mathematical notation to find the percentage of the larger portion after $x$ number of turns taking $75\%$ of the smaller portion and adding it to the larger portion?

Answer 1

You could write this as a recursive sequence:

$$\cases{a_0 = 0 \\ a_n = a_{n-1} + 0.75(1 - a_{n-1}) = 0.75 + 0.25 a_{n-1}}$$

This notation is kind of convenient because you can clearly see that the limit will eventually be the whole pie:

$$L = 0.75 + 0.25 L \implies 4 L = 3 + L \implies L = 1$$

Answer 2

Well, each step you have $\frac 14$ of what was previously left left. That is after $k$ steps you will have $(\frac 14)^k$ left.

You take $\frac 34$ of $(\frac 14)^k$ and add it to what you had.

So

Step 0: You have $\frac 34$ of the pie and $\frac 14$ left.

Step 1: You have $\frac 34 + \frac 34\times \frac 14$ and you have $(\frac 14)^2$ left.

Step 2: You have $\frac 34 + \frac 34 \times \frac 14 + \frac 34\times (\frac 14)^2$ and you have $(\frac 14)^3$ left .. and so on.

Step $k$: You will have $\frac 34 + \frac 34 \times \frac 14 + .... + \frac 34 \times (\frac 14)^k$ and you will $(\frac 14)^{k+1}$ of the pie left.

Thus $\sum\limits_{k=0}^n \frac 34\times (\frac 14)^k=\frac 34\sum\limits_{k=0}^n (\frac 14)^k$.

.... or if you prefer $3 \sum\limits_{k=1}^{n+1} (\frac 14)^k$

Now maybe you do, or do not, know how to express terms such a $1 + x + x^2 + x^3 + ........ + x^n$?