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Make a poster using equilateral triangles with sides 27, 9, 3 and 1 units assembled as stage 3 of the Von Koch fractal. Investigate areas & lengths when you repeat a process infinitely often.

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What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.

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When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

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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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What do you notice about these families of recurring decimals?

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The interval 0 - 1 is marked into halves, quarters, eighths ... etc. Vertical lines are drawn at these points, heights depending on positions. What happens as this process goes on indefinitely?

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

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What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?

This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.

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Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

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Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

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If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

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Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

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Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?

Challenge Level

What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?