AMC10 2008 10B, Problem 11

Suppose that $(u_n)$ is a sequence of real numbers satifying $u_{n+2}=2u_{n+1}+u_n$, and that $u_3=9$ and $u_6=128$. What is $u_5$?

Solution: If we plug in $n=4$, we get $128=2u_5+u_4.$ By plugging in $n=3$, we get $u_5=2u_4+9.$ This is a system of two equations with two unknowns. Multiplying the second equation by 2 and substituting into the first equation gives $128=5u_4+18 \Longrightarrow u_4=22$, therefore $u_5=\frac{128-22}{2}=53 \longrightarrow \textbf{\fbox{53}}$.

Answer: 53