Express $a_n$ and $b_n$

Question

Let $$a_{n+1} = \frac{a_n + b_n}2,\quad b_{n+1} = \frac{a_{n+1} + b_n}2.$$

Express $a_n$ and $b_n$ in terms of $a_1$, $b_1$ and $n$

Answer 1

Hint: $$\begin{pmatrix}a_{n+1}\\ b_{n+1}\end{pmatrix}=\begin{pmatrix}\frac12&\frac12\\ \frac14&\frac34\end{pmatrix}\begin{pmatrix}a_{n}\\ b_{n}\end{pmatrix} $$