Convergence of a sequence defined by a piecewise recurrence relation. 
Test for convergence of the sequence $\langle u_n\rangle,$ where $u_1=7$ and
$u_n=\begin{cases}5u_{n-1}+1,&\text{ if }u_{n-1}\text{ is odd}\\\\ \frac{u_{n-1}}2,&\text{ if }u_{n-1}\text{ is even}\end{cases}$       for all natural numbers $n\ge2.$

This is a piecewise defined sequence. If it was defined by a single case then I could have tried to express $u_n$ in terms of the base value $u_1$ and then I would apply the limit test. But since there are two cases so it has become difficult for me to draw a conclusion. How to approach this problem? Please give me some hints. Thanks in advance.

Note that all the terms of the sequence are integers. (How?)
Thus, if the sequence were to converge, it would have to be constant eventually. (Why?)
However, solving $u_{n+1} = u_n$ shows that this is not possible. (Take two separate cases according to $u_n$ being even or odd.)
Thus, the sequence diverges.
