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Why do this problem

In this problem a surprising number pattern can be explained by using an image. By understanding the significance of the image offered students are helped to perceive the general rule.


Possible approach

"Imagine a sequence of fractions where each one is half of the previous fraction."

Write these sums on the board and ask students to work them out:

$$\frac{1}{2} + \frac{1}{4} $$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8}$$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\frac{1}{16}$$

"What do you notice?"
"Do you think the pattern will continue?"
"How do you know?"

Offer students a chance to share their ideas, and then show the video, or recreate Charlie's diagram on the board. Perhaps ask students to recreate the diagram for themselves.

"How could you use the diagram to explain the patterns you have noticed?"
"Can you describe an expression for the sum $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\dots + \frac{1}{2^n}$?"
"Can you convince someone that your expression is correct for all values of $n$?"

Allow students some time to discuss in pairs, then bring the class together to share their insights. It is important to insist on clearly justified arguments that refer to the generality - a key question to ask is "How do you know it will always happen?".
The second part of the problem, looking at the sum of the sequence $1 + 2 + 4 + \dots + 2^n$, can be treated in the same way.

Possible extension

Diminishing Returns uses similar diagrams to explore other geometric series and can lead on to discussion of infinite sums.

Possible support

Students could start by working on Slick Summing, in which they are invited to explore the sums of simple arithmetic sequences.