Suppose that $(u_n)$ is a sequence of real numbers satisfying
\[u_{n+2}=2u_{n+1}+u_n\]and that $u_3=9$ and $u_6=128$. What is $u_5$?
This problem involves a recurrence relation, which is a way of defining a sequence based on previous terms. One way to approach this problem is to try to find a formula for the general term of the sequence, and then plug in $n=5$. Another way is to use the given information to work backwards and find $u_5$ directly. I will try the second way, since it might be faster and easier. To find $u_5$, I need to use the relation $u_{n+2}=2u_{n+1}+u_n$ with $n=4$. This means that $u_6=2u_5+u_4$. I know that $u_6=128$, so I can write $128=2u_5+u_4$. To find $u_4$, I need to use the relation again with $n=3$. This means that $u_5=2u_4+u_3$. I know that $u_3=9$, so I can write $u_5=2u_4+9$. Now I have two equations with two unknowns, $u_4$ and $u_5$. I can solve them by substituting one equation into the other. If I substitute $u_5=2u_4+9$ into $128=2u_5+u_4$, I get $128=2(2u_4+9)+u_4$. Simplifying this, I get $128=5u_4+18$. Subtracting $18$ from both sides, I get $110=5u_4$. Dividing both sides by $5$, I get $u_4=22$. Using this value of $u_4$, I can find $u_5$ by plugging it into $u_5=2u_4+9$. This gives me $u_5=2(22)+9$. Simplifying this, I get $u_5=53$. 