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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 $S_1, S_2, ... S_{r-1}$.
OK! Now Prove It
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
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?
NRICH is part of the family of activities in the Millennium Mathematics Project.