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Age 14 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

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.

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.

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?

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.

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

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

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