Predicting What Comes Next: Exploring Sequences and Progressions
Fractals, bounces and GP problems
Use geometric progressions to explore fractals and bouncing balls.
The Sierpiński triangle. Start with a black triangle. Join the midpoints of its sides and remove the middle triangle. Repeat with every black triangle that is left. It is a fractal: each part looks like the whole.
Two GPs in one picture:
- black triangles: 1, 3, 9, 27, 81, …: 3ⁿ at Stage n, a GP with r = 3;
- black area: each stage keeps 3/4 of it, so 1, 3/4, 9/16, …: (3/4)ⁿ, a GP with r = 3/4 that shrinks towards 0.
More and more triangles, but less and less area.
A small fix to the book. Its recursive rule 't₁ = 1, tₙ = 3tₙ₋₁' should start at Stage 0: t₀ = 1, tₙ = 3tₙ₋₁ for n ≥ 1. The Sierpiński carpet does the same with squares cut into nine: 8ⁿ squares, area (8/9)ⁿ.
A bouncing ball that rises to 3/4 of its height each time makes a GP. From 24 ft: 18, 13.5, 10.125, 7.59, 5.70, 4.27, 3.20, …. It first rises less than 1/6 of 24 ft (4 ft) on the 7th bounce.
Total distance travelled: count the first drop once, then each bounce twice (up and down).
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