Question

form an apollonian gasket
apollonian gasket
step 1: draw three circles that are tangent to each other, forming one triangle with curved sides, as shown in the figure.
step 2: draw a circle that is tangent to the three circles, forming three triangles with curved sides.
step 3: draw three circles, making each one tangent to the new circle and to two of the original circles, forming nine triangles with curved sides.
continue the steps by drawing circles that are tangent to three circles and form three triangles with curved sides.
how many curved triangles are in the ninth iteration?

Explanation:

Step1: Identify the pattern

We observe that the number of curved - sided triangles follows a geometric - sequence pattern. Initially, in step 1, we have 1 curved - sided triangle. In step 2, we have 1 + 3=4 curved - sided triangles. In step 3, we have 4 + 9 = 13 curved - sided triangles. The general formula for the number of curved - sided triangles $T_n$ at the $n$th iteration of constructing the Apollonian gasket is given by the recurrence relation. Let $T_1 = 1$. The number of new triangles added at the $k$th step is $3^{k - 1}\times3=3^{k}$.
The formula for the sum of a geometric series $S_n=\sum_{i = 0}^{n - 1}a\times r^{i}=\frac{a(1 - r^{n})}{1 - r}$ (where $a$ is the first term and $r$ is the common ratio).
The number of curved - sided triangles $T_n$ can be calculated as $T_n=\sum_{i = 0}^{n - 1}3^{i}$. Here, $a = 1$ and $r = 3$.

Step2: Apply the geometric - series formula

Using the formula for the sum of a geometric series $S_n=\frac{1\times(1 - 3^{n})}{1 - 3}=\frac{3^{n}-1}{2}$.

Step3: Calculate for $n = 9$

Substitute $n = 9$ into the formula $T_9=\frac{3^{9}-1}{2}$.
We know that $3^{9}=19683$. Then $T_9=\frac{19683 - 1}{2}=\frac{19682}{2}=9841$.

Answer:

9841