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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)?

Summing Geometric Progressions

Age 14 to 18 Challenge Level:
Watch the video below to see how Alison works out the sum of the first twenty terms of the sequence: $$2, 8, 32, 128, 512 ...$$

Can you adapt Alison's method to sum the following sequences?
  • $3, 9, 27, 81, 243 ...$ up to the 15th term
  • $5, 10, 20, 40, 80 ...$ up to the 12th term
  • $\sum_{i=1}^{20}(3 \times 2^{i-1})$
  • $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} ...$ up to the 10th term

Can you find an expression for the following sum up to the nth term? $$a + ar + ar^2 + ar^3 + ...$$