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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
Proof Sorter and article

Possible extension

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