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Find the sum of a finite geometric series.

The sides of an equilateral triangle measure 16 inches. The midpoints of the sides of the triangle are joined to form another equilateral triangle with sides that are half the length of the outer triangle. This process is continued until three triangles are inscribed in the first triangle. The sum of the perimeters of all four triangles is __________
inches.

12

To find the sum of a finite geometric series, we can use the formula:

S = a * (1 - r^n) / (1 - r)

Where:
- S represents the sum of the series
- a represents the first term of the series
- r represents the common ratio
- n represents the number of terms in the series

In this case, we have an equilateral triangle with sides measuring 16 inches.

The process described in the question involves creating smaller equilateral triangles inside the larger one, each with sides half the length of the previous triangle.

Since we are inscribing three triangles, we can determine the number of terms in the series as follows:

n = 3 + 1 = 4

Now let's calculate the sum of the perimeters of the triangles:

For the first equilateral triangle:
- a = 16 (since the side length is 16 inches)
- r = 1/2 (since each subsequent triangle has sides half the length of the previous triangle)

Using the formula, we get:

S = 16 * (1 - (1/2)^4) / (1 - 1/2)
= 16 * (1 - 1/16) / (1/2)
= 16 * (15/16) / (1/2)
= 15

Therefore, the sum of the perimeters of all four triangles is 15 inches.

12

a = 16*3 = 48

r = 1/2

So, what is S4?

16