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## 'Be Reasonable' printed from http://nrich.maths.org/

Why do this problem?
It is an exercise in proof by contradiction.

Possible approach
First discuss proof by contradiction so that students appreciate
how the logic of arguments by contradiction work. You can draw on
this article on

Proof by Contradiction.
Then discuss the proof that $\sqrt 2$ is irrational.

The students can work with the interactivity

Proof Sorter and perhaps some of them might read

this article which was written by two undergraduates.

Key Question

If the difference between$\sqrt 2$ and $\sqrt 3$ is an integer
multiple of the common difference in an arithmetic series, and the
difference between $\sqrt 3$ and $\sqrt 5$ is also an integer
multiple of that common difference, can you use these two facts to
write down two expressions, eliminate the unknown common difference
and then find an impossible relationship?

Possible support

Possible
extension

Can you prove that $\sqrt{1}$, $\sqrt{2}$ and $\sqrt{3}$
cannot be terms of ANY arithmetic progression?