Explanation on question about repetitive tiles 
The question is as follows: Consider this reptile (repetitive tile). 
How many divisions do we need to obtain:
A) At least $10^6$ smallest tiles?
B) At least $10^9$ total tiles?
I have the solutions for both of them. For A): 
$\log_{10} 4^{x} = \log_{10} 10^6$
$x\log_{10} 4 = 6$
$x = \frac{6}{\log_{10} 4} \implies x \approx 9.97$
For B):
$\frac{4^0 (1-4^{n+1})}{1-4} = 10^9$
$4^0(1-4^{n+1}) = -3(10^9)$
$3^{n+1} = 3(10^9)$
Can someone please explain what the process is for questions like this? I don't understand the solution. Specifically, where the 4 comes from in part A) and I don't understand how to even create the equation for part B). 
 A: Hint:
Each time you make a division you add three sub-tiles to every existing tile, so the number of tiles is multiplied by $4$ each time. This means that the number of smallest tiles after $n$ divisions is $4^n$. The total number of tiles after $n$ divisions is given by adding all previous tile values  $$1+4+4^2+\dots+4^n=\frac{4^{n+1}-1}{3}.$$
A: A) We start off with one small tile. After one division, there are four tiles: the one you started with, and the three you added to its corners.

In fact, after every division, you copy the figure you already have three times, so you end up with 4 times as many tiles as before the divisions. Therefore, the number of small tiles is $$ ST = 4^x$$ where $x$ is the number of divisions.
B) We start off with one small tile. After one division, there are four small tiles, and one 'large' tile of size 4, consisting of four small tiles itself. After the next division, there are sixteen small tiles, four tiles of size 4, and one tile of size 16. Following this process, it should be clear that the number of tiles after $x$ divisions is 
$$T = 1 + 4^1 + 4^2 + \ldots + 4^{x-1} + 4^x = \sum_{i = 0}^x 4^i$$
Now there is a neat formula for these kind of sums, which you have to know by heart:
$$\sum_{i = 0}^x a^i = \frac{a^{x+1}-1}{a-1}$$
so in particular here we have
$$\sum_{i = 0}^x 4^i = \frac{4^{x+1} - 1}{4 - 1}$$
