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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

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

The Root of the Problem

Age 14 to 18
Challenge Level

The Root of the Problem printable sheet

Alison has been exploring sums with surds. She used a spreadsheet to make columns for square roots, and then added together various combinations.

Here is one of the sums she worked out: $$\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + ... +\frac{1}{ \sqrt {99}+ \sqrt{100}}.$$

The answer surprised her!
Can you find a way to evaluate the sum without using a calculator or spreadsheet?

Click here for a hint:

When a fraction contains surds, we often choose to multiply the numerator and denominator by an expression that gets rid of any surds in the denominator.
Knowing that $(a+b)(a-b)=a^2-b^2$ might help.


Can you find other similar sums with surds that give whole number answers?