<s>[INST] $a_1 =2$ and $a_{n+1}= \frac{2a_n +3}{a_n +2}$ the recursive sequence , converges?, and if yes, show it to which converges I do not know what to do, I try showing the first elements to see if they were behaving in some way, but no
 [/INST] Another less known way of solving this particular form of recurrence relations.
The key is transform the non-linear recurrence to a linear one.
Notice $$\begin{bmatrix}a_{n+1}\\1\end{bmatrix}
\propto \begin{bmatrix}2 & 3\\1 & 2\end{bmatrix}
\begin{bmatrix}a_{n}\\1\end{bmatrix}, \forall n > 1
\quad\text{ and }\quad
\begin{bmatrix}a_{1}\\1\end{bmatrix} = \begin{bmatrix}2\\1\end{bmatrix} =
\begin{bmatrix}2 & 3\\1 & 2\end{bmatrix}
\begin{bmatrix}1\\0\end{bmatrix}$$
We have
$$\begin{bmatrix}a_{n}\\1\end{bmatrix}
\propto \begin{bmatrix}2 & 3\\1 & 2\end{bmatrix}^n
\begin{bmatrix}1\\0\end{bmatrix}, \forall n > 1$$
The matrix $\begin{bmatrix}2 & 3\\1 & 2\end{bmatrix}$ has eigenvalues $\lambda_{\pm}$ and corresponding eigenvectors $u_{\pm}$ given by
$$\lambda_{\pm} = 2\pm \sqrt{3}\quad\leftrightarrow\quad u_{\pm} = \begin{bmatrix}\pm\sqrt{3}\\1 \end{bmatrix}$$
Since $\begin{bmatrix}1\\0\end{bmatrix} \propto u_{+} - u_{-}$, we have
$\begin{bmatrix}a_{n}\\1\end{bmatrix} \propto \lambda_{+}^n u_{+} - \lambda_{-}^n u_{+}$. Together with $\lambda_{+}\lambda_{-} = 1$ and $|\lambda_{-}| < 1$, we find
$$a_n = \sqrt{3}\frac{\lambda_{+}^n - \lambda_{-}^n}{\lambda_{+}^n - \lambda_{-}^n} =
\sqrt{3}\frac{1 + \lambda_{-}^{2n}}{1 - \lambda_{-}^{2n}}
= \sqrt{3}\frac{1 + (2-\sqrt{3})^{2n}}{1 - (2-\sqrt{3})^{2n}}
\to \sqrt{3}\quad\text{ as } n \to \infty$$
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