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

Blue: 1/16 square units.

Green: 1/64 square units.

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

If the pattern continues indefinitely, the sum of the area is 1/3 square units.

Explanation:

I got it all right.

Answer 2

The correct answers are Blue:
`(1)/(16)` square units ; Green:
`(1)/(64)` square units; and Yes the sum converges.

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
`(1)/(4)`.

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

Area of the orange triangle is given by 1 ×
`(1)/(4)` =
`(1)/(4)` square units.

Area of the blue triangle is given by
`(1)/(4)` ×
`(1)/(4)` =
`(1)/(16)` square units.

Area of the green triangle is given by
`(1)/(16)` ×
`(1)/(4)` =
`(1)/(64)` square units.

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