Problem source

The sequence $u_n$ ($n= 1, 2, \ldots$) satisfies the recurrence relation 
\[ 
u_{n+2}= \frac{u_{n+1}}{u_n}(ku_n-u_{n+1})  
\] 
where  $k$ is a constant. 
 
If $u_1=a$ and $u_2=b\,$,  
where $a$ and  $b$ are non-zero and $b \ne ka\,$, prove by induction that 
\[ 
u_{2n}=\Big(\frac b a \Big) u_{2n-1} 
\] 
\[ 
u_{2n+1}= c u_{2n} 
\] 
for $n \ge 1$,  
where $c$ is a constant to be found in terms of $k$, $a$ and $b$. 
Hence express $u_{2n}$ and $u_{2n-1}$ in terms of $a$, $b$, $c$ and $n$. 
 
Find conditions on $a$, $b$ and $k$ in the three cases: 
\begin{questionparts}
\item the sequence $u_n$ is geometric;
\item $u_n$ has period 2; 
\item the sequence $u_n$  has period 4. 
\end{questionparts}