Question 3
The sequence $(x_n)$ is defined by
$$x_0=2,\qquad x_{n+1}=\left[\sqrt{2x_n(x_n+1)}\right]\quad(n\in\mathbb N^*).$$
Prove that
$$x_{2n}=2^n+[2^n\sqrt2],\qquad x_{2n+1}=2^{n+1}+[2^n\sqrt2]$$
for every $n\in\mathbb N^*$.
$$x_0=2,\qquad x_{n+1}=\left[\sqrt{2x_n(x_n+1)}\right]\quad(n\in\mathbb N^*).$$
Prove that
$$x_{2n}=2^n+[2^n\sqrt2],\qquad x_{2n+1}=2^{n+1}+[2^n\sqrt2]$$
for every $n\in\mathbb N^*$.