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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 $S_1, S_2, ... S_{r-1}$.

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Degree Ceremony

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

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OK! Now Prove It

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

Reciprocal Triangles

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

The algebraic expression for $r$th triangular number is

$$T_r = \frac{1}{2} r(r+1) $$

 

The expression that you are trying to evaluate is $$\sum_{r=1}^{n} \frac{1}{T_r} = \frac{1}{T_1} + \frac{1}{T_2} + \frac{1}{T_3} + ... + \frac{1}{T_n} \cong 2 $$