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Lesson 7 of 8

Predicting What Comes Next: Exploring Sequences and Progressions · Lesson 7 of 8

Geometric Progressions in Fractals and Bounces

“Fractals keep repeating while bounces keep shrinking—both know how to make a pattern last.”

Learning Objectives

• Apply GPs to fractal patterns. • Understand self-similarity in the Sierpiński triangle. • Model increasing counts and decreasing areas with GPs. • Interpret GP graphs. • Apply geometric decay to a bouncing ball.

Geometric progressions appear when a quantity changes by the same multiplication factor from one stage to the next. Instead of adding or subtracting a fixed amount each time, we repeatedly multiply by the same number. This creates a pattern in which the values can grow very quickly or become smaller and smaller, depending on the common multiplier.

These patterns are useful for describing many situations in mathematics and the real world. Fractals, for example, are built by repeating the same geometric rule at smaller and smaller scales, while a bouncing object may reach a fixed fraction of its previous height after each bounce. In both cases, repeated multiplication creates a geometric pattern that can be studied using a geometric progression.

The Sierpiński Triangle

Begin with one filled equilateral triangle. Divide it into four equal smaller triangles and remove the middle one. Repeat the same process on each remaining filled triangle. The pattern repeats at smaller and smaller scales.

Definition
Fractal

A shape or pattern that repeats similar structure at different scales.

Stages of a Sierpiński triangle Stages zero through three of a Sierpiński triangle. At every new stage, the central inverted triangle is removed from each remaining solid triangle. Stage 0 Stage 1 Stage 2 Stage 3 Solid triangle Remove the centre Repeat in each triangle Repeat once more 1 solid triangle 3 solid triangles 9 solid triangles 27 solid triangles At every stage, each solid triangle becomes 3 smaller triangles
Stages of a Sierpiński triangle

The number of black triangles is 1,3,9,27,… . Each stage has three times as many black triangles as the previous one.

Number of black triangles at stage nLaTeX

But the total black area behaves in the opposite way. At every stage, only three of four equal sub-triangles remain, so the area is repeatedly multiplied by 3/4.

Black area at stage nLaTeX
A surprising contrast

The number of pieces can grow rapidly while the total shaded area shrinks. One GP has ratio 3; another has ratio 3/4.

Visualising a GP

For 3,6,12,24,48,… the points (1,3),(2,6),(3,12),(4,24),(5,48) do not lie on one straight line. The amount added is not constant; it keeps getting larger because each value doubles.

Graph of the GP 3, 6, 12, 24, 48 Discrete points representing the geometric progression 3, 6, 12, 24, 48. The points form an increasingly steep curve-like pattern. Graph of the GP 3, 6, 12, 24, 48 Each term is twice the previous term, so the graph becomes increasingly steep aₙ = 3 × 2ⁿ⁻¹ 1 2 3 4 5 0 12 24 36 48 Term number, n Term value, aₙ (1, 3) (2, 6) (3, 12) (4, 24) (5, 48) Each value doubles: 3 → 6 → 12 → 24 → 48 The points are discrete, but their pattern follows an increasingly steep curve
Graph of GP 3,6,12,24,48

Bouncing Ball as Geometric Decay

Suppose a ball rises to 3/4 of its previous height after each bounce. The heights form a GP with ratio 3/4, so every bounce is lower than the one before it.

Bouncing Ball

Problem
A ball is dropped from 24 ft and each rebound reaches 3/4 of the previous height. Write the first five rebound heights.

  1. 1.First rebound: 24×3/4=18 ft.
  2. 2.Second: 18×3/4=13.5 ft.
  3. 3.Third: 13.5×3/4=10.125 ft.
  4. 4.Fourth: 10.125×3/4=7.59375 ft.
  5. 5.Fifth: 7.59375×3/4≈5.695 ft.

This is geometric decay: multiplying repeatedly by a positive number less than 1 makes terms smaller and smaller.

Practice Problems

Practice Problems
  1. A fractal has 1 piece at Stage 0 and each piece becomes 4 pieces at the next stage. Write the first six counts and an explicit rule.
  2. A shaded area starts at 1 and becomes 2/3 of the previous area each stage. Write the first five terms.
  3. A ball rebounds to 60% of its previous height. If the first rebound is 48 m, find the next four rebound heights.
  4. Explain why 3,6,12,24,… does not produce a straight-line pattern when plotted against term number.
  5. Compare the long-term behaviour of ratios 2 and 1/2 in geometric progressions.

Key Takeaways

Key Takeaways

• Fractals often create geometric progressions. • Repeated multiplication by r>1 gives rapid growth. • Repeated multiplication by 0<r<1 gives decay. • A Sierpiński triangle gives both a growing count GP and shrinking area GP. • Bouncing heights are a natural example of geometric decay.

Coming Next

Next, we summarise the complete chapter and practise choosing between explicit rules, recursive rules, APs and GPs.

Previous · Lesson 6

Geometric Progressions

Next · Lesson 8

Chapter Summary and Practice