> 
> Question: Let $a\_0,b\_0,c\_0$ be real numbers.
> Define the sequences $(a\_n)\_n,(b\_n)\_n, (c\_n)\_n$ recursively by
> $$a\_{n+1} = \frac{1}{2}(a\_n+b\_n), \quad b\_{n+1} = \frac{1}{2}(b\_n+c\_n) \quad \text{and}\quad c\_{n+1} = \frac{1}{2}(c\_n+a\_n).$$
> Prove that the sequences are convergent and find their limits.
> 
> 
> 

My attempt: I try to find the closed-form solution for $a\_n,b\_n,c\_n$.
Note that
$$
\begin{pmatrix}
a\_{n+1} \\
b\_{n+1} \\
c\_{n+1} \\
\end{pmatrix} = \begin{pmatrix}
\frac{1}{2} & \frac{1}{2} & 0 \\
0 & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & 0 & \frac{1}{2}
\end{pmatrix} \begin{pmatrix}
a\_n \\
b\_n \\
c\_n
\end{pmatrix} = A\begin{pmatrix}
a\_n \\
b\_n \\
c\_n
\end{pmatrix}.
$$
Clearly
$$
\begin{pmatrix}
a\_{n} \\
b\_{n} \\
c\_{n} \\
\end{pmatrix} = A^{n}\begin{pmatrix}
a\_0 \\
b\_0 \\
c\_0 \\
\end{pmatrix}.
$$
To compute $A^n$, I try to diagonalize $A$ over complex numbers.
From [Wolfram alpha](https://www.wolframalpha.com/input/?i=%7B%20%7B0.5%2C0.5%2C0%7D%2C%20%7B0%2C0.5%2C0.5%7D%2C%20%7B0.5%2C0%2C0.5%7D%20%7D), $A$ has eigenvalues $1, \frac{1}{4}(1+i\sqrt3)$ and $\frac{1}{4}(1-i\sqrt3)$.
Therefore,
$$
\begin{pmatrix}
a\_{n} \\
b\_{n} \\
c\_{n} \\
\end{pmatrix} = X \begin{pmatrix}
1 & 0 & 0 \\
0 & \left( \frac14 (1+i\sqrt 3) \right)^n & 0 \\
0 & 0 & \left( \frac14 (1-i\sqrt 3) \right)^n
\end{pmatrix} X^{-1} \begin{pmatrix}
a\_0 \\
b\_0 \\
c\_0 \\
\end{pmatrix}
$$
for some invertible matrix $X$ (in fact, columns of $X$ are eigenvectors of $A$).
However, I stuck at here.
I do not know how to proceed to show that the sequences $(a\_n), (b\_n), (c\_n)$ convergent.

It would be good if someone can solve this problem by continuing my method above.

---

EDIT: After reading comments below, I attempted the following.
From [Wolfram alpha](https://www.wolframalpha.com/input/?i=Limit%5BMatrixPower%5B%7B%20%7B0.5%2C0.5%2C0%7D%2C%20%7B0%2C0.5%2C0.5%7D%2C%20%7B0.5%2C0%2C0.5%7D%20%7D%2Cn%5D%2Cn%2CInfinity%5D), we have
$$\lim\_{n\to\infty} A^n = \frac13 J\_3$$
where $J\_3$ is the $3\times 3$ matrix with all entries $1$.
Therefore,
$$\lim\_{n\to\infty} \begin{pmatrix}
a\_{n} \\
b\_{n} \\
c\_{n} \\
\end{pmatrix} = \frac13 J\_3 \begin{pmatrix}
a\_{0} \\
b\_{0} \\
c\_{0} \\
\end{pmatrix} = \frac13 \begin{pmatrix}
a\_{0} + b\_{0} + c\_{0} \\
a\_{0} + b\_{0} + c\_{0} \\
a\_{0} + b\_{0} + c\_{0}
\end{pmatrix}.$$
Hence, the sequences $(a\_n),(b\_),(c\_n)$ are convergent and their limits are $\frac13 (a\_0 + b\_0 + c\_0)$.