Does the $5x + 1$ sequence for 7 reach a power of 2 or does it get stuck in a period? This is much like the $3x + 1$ iteration, except that if $x$ is odd, you do $5x + 1$ [and $\frac{x}{2}$ if $x$ is even]. If $x = 7$, then we have 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, 458, 229, ...
I've iterated this twenty thousand times and found no power of 2. It's also possible that there is a period but it's so large that I'm not spotting it. And it's also possible that I have made a mistake somewhere along the way.
Surely someone else has also calculated this, even though $5x + 1$ is not as famous as $3x + 1$. I have tried other starting $x$ and seen that they quickly reach a period. Has anyone determined what happens with $x = 7$?
 A: In Lagarias' paper from 1985, he mentions (on the bottom of p. 12) this (5x+1) variation.  He says that stochastic models predict that almost all orbits escape to infinity.  However, this is a heuristic only; it was not proved (in 1985) that even a single orbit escapes to infinity.  He calls this open problem (C3), on p. 22.  This appears to remain open circa 2006, for here is a paper by Volkov, behind a paywall unfortunately, that continues study of this problem.  Volkov gives three cycles, and computational (and heuristic) evidence that all other orbits diverge.
A: This is giving me some deja vu from centuries ago, but I don't know if it's because I actually found the answer and then forgot it. So I'm not sure if it reaches a period that has evaded detection by  you and your colleages. I can tell you this much: it sure doesn't hit a power of $2$. Consider the sequence modulo $8$:
$$7, 4, 2, 1, 6, 7, 4, 2, 5, 2, 1, 6, 7, 4, 2, 1, \ldots$$
In order for this to hit a power of $2$, it would have to hit $4$ itself.
A: this is a copy of an older answer on math.SE, see the duplicate question here. Your number has the number $35$ as pre-precedessor and its route can be seen on the pictures below. 
Here are some pictures for your/our intuition. I graphed the trajectories for initial values $x=5,15,25,...$ for the first $256$ steps of $x_{k+1}=(5x_k+1)/2^A$.
To get the curves to a meaningfully visual interval I show logarithmic scales. The pictures show how most trajectories begin to diverge (not really a safe indication of what characteristic the infinite curves really have) but some show cycling already at early iteration indexes $k$ .    
I find $2$ cycles besides the "trivial" one.

$x=5,15,25,35,...,95$ detail of the first few iterations . At the bottom we see the "trivial" cyle (brown curve):


$x=5,15,25,35,...,95$ first $2^8 = 256$ iterations. At later iteration-indexes $k$ a first "non-trivial" cycle occurs (red line):


$x=105,115,125,135,...,195$ first $2^8 = 256$ iterations .


$x=205,215,225,235,...,295$ first $2^8 = 256$ iterations . Here a second "non-trivial" cycle becomes visible:


$x=205,215,225,235,...,295$ first $2^{11} = 2048$ iterations
It seems really that all trajectories which are divergent up to iteration $k=256$ are also divergent up to iteration $k=2048$ . In general: I doubt that there are "later" cycles:


