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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)?

Golden Fibs

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

This is straightforward but care must be taken to explain the significance of both roots of the quadratic equation. Also you must prove both the 'if' and the 'only if', that is, for general Fibonacci sequences, 'if the first two terms are in the golden ratio then the sequence is geometric' and 'if the sequence is geometric then the ratio of successive terms is the golden ratio'.

Induction is required