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Geometric Sequences

Source: Fractal Art - Instants/iStock/Getty Images 

Familiar Sequences

 

 

 

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Lesson Goals

  • Identify a geometric sequence.
  • Determine a recursion formula for a geometric sequence.
  • Determine and apply the formula for the general term of a geometric sequence.
  • Solve problems involving geometric sequences.

Math in Action

A fractal is a geometric figure in which the same pattern appears at different scales — that is, the whole object is similar or approximately similar to parts of itself.

Fractals appear often in nature, such as in the branching patterns of trees, lightning and rivers, the spiral patterns of a hurricane or seashells, and the shape of a snowflake.

Two well-known geometric fractals are the Sierpinski triangle and the Koch snowflake. Both of these fractals are constructed by repeating the same process over and over, producing a series of figures that increase in complexity with each iteration.

Try This

Here are the first four iterations of the Sierpinski triangle.

First triangle is completely dark, second triangle is divided into four triangles where the middle one is light, third triangle all the dark triangles from the second triangle are divided into four triangles again where the middle one is light, and so on with the fourth triangle.

Source: Sierpinski Triangle - alejomiranda/iStock/Getty Images

The first iteration can be seen as an equilateral triangle that is \(100 \%\) dark. In subsequent iterations, the large triangle is covered by smaller, white, equilateral triangles. The percentage of the large triangle that is dark decreases from iteration to iteration.

After the \(\)11th iteration, what percentage of the Sierpinski triangle is dark?