Problem 2. Determine the general term of the sequence $\left(x_{n}\right)_{n \geq 1}$ defined by:

$$
x_{1}=1, x_{2}=2 \text { and } x_{n+2}=\frac{x_{n+1}^{2}}{\left[\sqrt{x_{n}^{2}+x_{n+1}}\right]}-\frac{1}{n}, n \geq 1
$$

Marius Perianu, Slatina
## Solution and marking scheme

$x_{3}=3$

We will prove by mathematical induction the proposition $P(n): a_{n}=n$, for all $n \geq 1$

$P(1), P(2)$ are true

$\left[\sqrt{k^{2}+k+1}\right]=k$

$x_{k+2}=k+2$, so $P(k+2)$ is true; therefore $a_{n}=n$, for all $n \geq 1$