MathNet 2010 — Part 02 Question 48

Let $x_1$ and $x_2$ be real numbers and $0 \lt p \lt 1$. Define $x_n = p x_{n-1} + (1-p) x_{n-2}$ for $n = 3, 4, \dots$. Show that the sequence $(x_n)$ converges and determine $\lim_{n \to \infty} x_n$.