## Exercise 8. Calculate

$$
\sqrt{7+\sqrt{1+\sqrt{7+\sqrt{1+\sqrt{7+\sqrt{1+\sqrt{7+\sqrt{3+\sqrt{1}}}}}}}}}
$$

Only a numerical answer is expected here.
Solution to exercise 8 We notice the following chain of simplifications: $\sqrt{3+\sqrt{1}}=2$ then $\sqrt{7+\sqrt{3+\sqrt{1}}}=\sqrt{7+2}=3$ then $\sqrt{1+\sqrt{7+\sqrt{3+\sqrt{1}}}}=\sqrt{1+3}=2$, and so on...

Step by step, we deduce that any expression of the type $\sqrt{1+\sqrt{7+\sqrt{1+\sqrt{\cdots}}}}$ equals 2 and that any expression of the type $\sqrt{7+\sqrt{1+\sqrt{7+\sqrt{\cdots}}}}$ equals 3, regardless of the number of square roots, provided that $4=3+\sqrt{1}$ is under the last radical.

Since a 7 is found in the sum under the first radical, the answer is therefore 3.

Comment from the graders: Overall, students were not misled by the nesting of all the square roots and performed very well on this exercise. However, it is worth noting that many students wrote down the result without any justification: as mentioned in the instructions, this risks a direct 0 in case of an incorrect answer, it is better to justify the answer to some extent to avoid disappointment in case of a calculation error.