Question

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? ______

Answer 1

Answer:

The correct answers are Blue: $\frac{1}{16}$ square units ; Green:  $\frac{1}{64}$ 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 $\frac{1}{4}$.

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

Area of the orange triangle is given by 1 × $\frac{1}{4}$ = $\frac{1}{4}$ square units.

Area of the blue triangle is given by $\frac{1}{4}$ × $\frac{1}{4}$ = $\frac{1}{16}$ square units.

Area of the green triangle is given by $\frac{1}{16}$ × $\frac{1}{4}$ = $\frac{1}{64}$ square units.

If the pattern continues definitely then it becomes an infinite geometric progression series with common ratio $\frac{1}{4}$. And thus the sum of area coverges as the ratio lies between -1 and 1.