The area of the shaded triangles in the fractal shown form a geometric sequence. The area of the largest triangle (not shaded) is 1 square unit. Find the areas of these shaded triangles.

Orange: 1/4 square units

Blue: square units

Green: square units

If the pattern continues indefinitely, does the sum of the areas converge?

The correct answers are Blue: square units ; Green:  square units; and Yes the sum converges.

Step-by-step explanation:

Area of the largest triangle is 1 square unit.

The area of the triangle which are colored and follow a geometric sequence with the common ratio being .

Area of the not shaded triangle is given by 1 square unit.

Area of the orange triangle is given by 1 × = square units.

Area of the blue triangle is given by × = square units.

Area of the green triangle is given by × = square units.

If the pattern continues definitely then it becomes an infinite geometric progression series with common ratio . And thus the sum of area coverges as the ratio lies between -1 and 1.

after simplifying it i got: -36.25 as my answer. hope that !

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