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Converging Product

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

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Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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Binary Squares

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Sixty-seven Squared

Stage: 5 Challenge Level: Challenge Level:1

Why do this problem?

An excellent exercise in using geometric series with a fun way into the problem through experimenting with numbers and making conjectures.

Possible approach

Let the students experiment with the numbers until they spot patterns and make conjectures. If they don't see how to prove the general result you may have to ask the 'key question' and suggest using the sum of a geometric series.

Key questions

Can you write the number (k sixes followed by a 7) in terms of powers of 10 which will work for all k?

Possible extension

Try the problem Clickety Click and All the Sixes