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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

Why do this problem

This problem offers students an opportunity to practise manipulating surds in the denominator, and highlights the importance of not rounding off prematurely, as by keeping surds in the calculation and simplifying as much as possible, a pleasing answer emerges that might be hidden if students used a calculator and rounded their answers along the way.


Possible approach

Invite students to use spreadsheets to sum parts of the sequence: $$\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + ... +\frac{1}{ \sqrt {99}+ \sqrt{100}}.$$

We hope students will be surprised when they notice that at various points in the sequence, the sum is a whole number, and that they will conjecture about when this happens and wish to explain it. They may need reminding about techniques to rationalise the denominator.



Key questions

For which values of $n$ does the series give whole numbers?
Why might that be?
Can we express $\frac{1}{\sqrt{n}+\sqrt{n+1}}$ in a way that the surds are in the numerator rather than the denominator?


Possible support

Students could start by finding an expression for $\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}}$ and then add subsequent terms.

Possible extension

Irrational Arithmagons and Ab Surd Ity are both challenging problems involving the manipulation of surds.