<s>[INST] Find expression of $c_n$, where $c_n = a_n + b_n$ Given the recurrence relation $a_{n+2} = 3a_{n+1} + 6a_n$ and $b_{n+2} = b_{n+1} + b_n$ I am supposed to find an expression of the recurrence relation for $c_n := a_n + b_n$. I tried to find some form of linear dependence to obtain a recurrence relation for $c_n$ but this did not let me finish.
Appreciate your help!
 [/INST] Using that $\,a_{n+2} - 3a_{n+1} - 6a_n = 0\,$ and $\,b_{n+2} = b_{n+1} + b_n\,$:
$$
\require{cancel}
\begin{align}
c_{n+2}-3c_{n+1}-6c_n &= \cancel{(a_{n+2}-3a_{n+1}-6a_n)} + (b_{n+2}-3b_{n+1}-6b_n)
\\ &= -2b_{n+1} - 5 b_n
\end{align}
$$
Then, using that $\,b_{n+2} - b_{n+1} - b_n = 0\,$:
$$
\begin{align}
(c_{n+4} - 3c_{n+3}-6c_{n+2}&) - (c_{n+3}-3c_{n+2}-6c_{n+1}) - (c_{n+2}-3c_{n+1}-6c_n)
\\ &=\; -2\cancel{(b_{n+3}-b_{n+2}-b_{n+1})} - 5\bcancel{(b_{n+2}-b_{n+1}-b_{n})}
\\ &= 0
\end{align}
$$
$$
\iff c_{n+4} - 4c_{n+3}-4c_{n+2}+9 c_{n+1}+ 6 c_n = 0
$$
</s>