Finding the limit of the sequence $T_1=0,T_2=1, T_n=\frac{T_{n-1}+T_{n-2}}{2}$ Given that $T_1=0$, $T_2=1$ and $T_n=\frac{T_{n-1}+T_{n-2}}{2}$, show that the sequence converges to $\frac{2}{3}$.

Start from the matrix representation of the problem
$$\begin{bmatrix}
T_{n} & T_{n-1} 
\end{bmatrix}=\begin{bmatrix}
T_{n-1} & T_{n-2} 
\end{bmatrix}
\begin{bmatrix}
 \frac{1}{2}& 1\\ 
 \frac{1}{2}& 0
\end{bmatrix}$$
$$\begin{bmatrix}
T_{2} & T_{1} 
\end{bmatrix}=\begin{bmatrix}
1 & 0 
\end{bmatrix}$$
Chaining it all you have
$$\begin{bmatrix}
T_{n} & T_{n-1} 
\end{bmatrix}=\begin{bmatrix}
T_{2} & T_{1} 
\end{bmatrix}
\begin{bmatrix}
 \frac{1}{2}& 1\\ 
 \frac{1}{2}& 0
\end{bmatrix}^{n-2}$$
So it all comes down to finding
$$\lim\limits_{n \to \infty}
\begin{bmatrix}
 \frac{1}{2}& 1\\ 
 \frac{1}{2}& 0
\end{bmatrix}^{n}$$
Now this part is really classical and you just find eigenvalues of the matrix $\lambda_{1}=-\frac{1}{2}$,$\lambda_{2}=1$ and diagonalize
$$
\begin{bmatrix}
 \frac{1}{2}& 1\\ 
 \frac{1}{2}& 0
\end{bmatrix}=\begin{bmatrix}
 -1 & 2\\ 
  1 & 1
\end{bmatrix}
\begin{bmatrix}
 -\frac{1}{2}& 0\\ 
 0 & 1
\end{bmatrix}
\begin{bmatrix}
 -\frac{1}{3} & \frac{2}{3}\\ 
 \frac{1}{3} & \frac{1}{3}
\end{bmatrix}$$
to have
$$\lim\limits_{n \to \infty}
\begin{bmatrix}
 -1 & 2\\ 
  1 & 1
\end{bmatrix}
\begin{bmatrix}
 -\frac{1}{2}& 0\\ 
 0 & 1
\end{bmatrix}^{n}
\begin{bmatrix}
 -\frac{1}{3} & \frac{2}{3}\\ 
 \frac{1}{3} & \frac{1}{3}
\end{bmatrix}=\lim\limits_{n \to \infty}
\begin{bmatrix}
 -1 & 2\\ 
  1 & 1
\end{bmatrix}
\begin{bmatrix}
 (-\frac{1}{2})^{n}& 0\\ 
 0 & 1
\end{bmatrix}
\begin{bmatrix}
 -\frac{1}{3} & \frac{2}{3}\\ 
 \frac{1}{3} & \frac{1}{3}
\end{bmatrix}=$$
$$\begin{bmatrix}
 -1 & 2\\ 
  1 & 1
\end{bmatrix}
\begin{bmatrix}
 0 & 0\\ 
 0 & 1
\end{bmatrix}
\begin{bmatrix}
 -\frac{1}{3} & \frac{2}{3}\\ 
 \frac{1}{3} & \frac{1}{3}
\end{bmatrix}=\frac{1}{3}\begin{bmatrix}
 2 & 2\\ 
 1 & 1
\end{bmatrix}$$
This is giving
$$\lim\limits_{n \to \infty}\begin{bmatrix}
1 & 0 
\end{bmatrix}
\begin{bmatrix}
 \frac{1}{2}& 1\\ 
 \frac{1}{2}& 0
\end{bmatrix}^{n-2}=\begin{bmatrix}
1 & 0 
\end{bmatrix}\frac{1}{3}\begin{bmatrix}
 2 & 2\\ 
 1 & 1
\end{bmatrix}=\begin{bmatrix}
 \frac{2}{3} & \frac{2}{3}
\end{bmatrix}$$
which means that the series converges to $\frac{2}{3}$
