Limit of a sequence of averaged numbers?

Question

Let $a_0 = 0$, $a_1 = 1$, and $a_n = \frac{a_{n-1}+a_{n-2}}{2}$ for all $n \ge 2$. Consider $\lim \limits_{n \to \infty} a_n$.

Using a quick python script I found that for large $n$ $a_n$ tends to $\frac{2}{3}$. How do I prove this result?

Answer 1

We have $$-2(a_n-a_{n-1})=a_{n-1}-a_{n-2}.$$ So $$a_n-a_{n-1}={(-\frac{1}{2})}^{n-1}.$$ So $$a_n=\sum_{i=1}^n{(-\frac{1}{2})}^{i-1}.$$ Now you know how to compute the limit.