Question

form an apollonian gasket
apollonian gasket
an apollonian gasket is formed by the following steps:
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

The number of curved - triangles follows a geometric - sequence pattern. Initially, there is 1 triangle. After the first step of adding a circle, there are 3 triangles. After the second step, there are 9 triangles. The common ratio \(r = 3\).

Step2: Use the geometric - sequence formula

The formula for the \(n\)th term of a geometric sequence is \(a_n=a_1r^{n - 1}\), where \(a_1\) is the first - term, \(r\) is the common ratio, and \(n\) is the number of the term. Here, \(a_1 = 1\), \(r = 3\), and \(n=9\).

Step3: Calculate the number of curved triangles

Substitute the values into the formula: \(a_9=1\times3^{9 - 1}=3^8 = 6561\).

Answer:

6561