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Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

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? $$a + ar + ar^2 + ar^3 + ...$$