The general thrust of this question is how big should I expect the territory of colonization to be?
The factors that I know...
You have 5 colony ships with 2 million colonists each that set up colonies on 5 planets. They are cut off from Earth and have to start from scratch.
Edit: Sorry - 2 million is just the initial numbers of the first colonies. Colony ships after that hold 1600 to 2200. Also they live to around 120 years old.
I currently assume that there is 245 years between this initial colonization and these colonies start to send out ne colonies.
When they set up their colony they have the ships and can build a "jump gate". These ships can travel at roughly 30c normally, and 300c assuming their is a "jump gate" at both the start and destination, which means they need to get to any place first and then they can travel 10x quicker.
Let's say this speed holds true from year 0 (initial colonization date) to year 182. After that they can travel at 40c or 400c.
Each new colony they set up, they will start building a new colony ship and launch it when they reach a population of 2 billion. Also, these colony ships serve as a space elevator/space station when they arrive.
What i need to know is, how by this sphere of colonization will be in year 513. Assuming a continuous travel, they reach ~10,000 light years out, but that doesn't seem right that they'd colonize all that,
Also assume they have infinite energy and infinite computer processing, but need to get the resources from the planets. The stations are used as habitats too, but each person has a plot of land that they exclusively own.
So, how small/big can I arguably make this sphere of territory given these "facts"