Does $x_{n+2} = (x_{n+1} + x_{n})/2$ converge? Let $0 < a < b < \infty$. Define $x_1=a$, $x_2=b$, and $x_{n+2} = \frac{x_n + x_{n+1}}{2}$ for $n \geq 1$. Does $\{x_n\}$ converge? If so, to what limit?

$$x_3 = \dfrac{a+b}2$$ $$x_4 = \dfrac{a+3b}4$$ $$x_5 = \dfrac{3a+5b}8$$ $$x_6 = \dfrac{5a+11b}{16}$$ $$x_7 = \dfrac{11a+21b}{32}$$
In general, (by induction) $$x_n = \dfrac{J_{n-2} a + J_{n-1}b}{2^{n-2}}$$ where $J_n$'s are Jacobsthal sequence i.e. they satisfy the recurrence
$$J_{n+1} = J_n + 2J_{n-1}$$ with $J_0 = 0$ and $J_1 = 1$. Once we have this recurrence, it is not hard (by induction) to find that
$$J_n = \dfrac{2^n - (-1)^n}3$$
Hence, $$x_n = \dfrac{(2^{n-2} - (-1)^{n-2})a + (2^{n-1} - (-1)^{n-1})b}{3 \cdot 2^{n-2}} \to \dfrac{a + 2b}{3}$$
