Given:
The sequence $(u_{n})$ with $u_{n+2}=4 \cdot u_{n+1}-u_{n}$
and $u_{4}=194$
with $u_{1}<u_{2}$ , $u_{i} \in \mathbb{N}_{0}$
To solve: $u_{1},u_{2},u_{3},u_{5}$
I tried solving it by replacing $u_{3}$ in $u_{4}= 4u_{3} - u_{2}$ with $4u_{2} - u_{1}$
Like this I got $194 = 15u_{2} - 4u_{1}$
But it doesn't really get me anywhere. I tried the same logic till $u_{5}$, but no success. This way I thought I would get an equation somehow to solve for $u_{1}$.
I think I'm overlooking something or not even getting close.
Now, $$u_2=\frac{4u_1+194}{15}>u_1,$$ which gives $$1\leq u_1\leq17$$ and only $u_1=4$ is valid.
Can you end it now?