1001math: Geometric progressions

Geometric sequences

Geometric progressions are often presented purely as a sequence of numbers. The ‘Geometric progressions’ resource from 1001math.com uses graphical representations to bring the topic to life.

The puzzles present geometric progressions where the common ratio is a fraction, shown as an area on a diagram, with the sum of the series being the fraction of the shape that is shaded. While it is possible to arrive at the sums using the formula for summing a geometric progression, each is accessible using logic.

The challenge is to find the fraction of the shape that is shaded. In this square each quarter has the same area, so we only need to focus on one of them. This is the upper right quarter.

Top right corner of sequence

This can be thought of as a series of L-shaped pieces that get smaller and smaller. While the Ls are all different in size, they each have the same fraction shaded:

Geometric sequences top right corner

Since one-third of each L is shaded, and the whole figure is composed of these Ls, then one-third of the big square is shaded.

Subtle alterations can make a big difference to the difficulty of the problem. Consider the diagram below:

Second geometric sequence

As previously, it is helpful to consider one quarter of the shape.

Second geometric sequence split

The largest L is three-quarters of the shape and is two-thirds shaded. The remainder of the shape is one-quarter and as in the previous example one-third of it is shaded.

The calculation of the area becomes

Calculation

We have recently added the complete collection of resources from Sian Zelbo’s ‘1001math’ website, the materials include investigations, puzzles, and other enrichment activities.   

You can also join us at the National STEM Centre in York to experience ideas for teaching through understanding on the following Enthuse funded courses: more information can be found here

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Subject(s)Mathematics
Age11-14, 14-16
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