FTL (Faster than Light) travel exists,taking around 88 hours 48 seconds to travel 1 light year. FTL communications also exists and takes only 53.28 seconds to travel 1 LY (Light Year). Jumps longer than 10 LY at a time have not been tried,with the safe range being considered around 5LY due to the great stresses on crew and to a lesser extent the ship during subspace travel. This would be for the crew an 18 days and 12 hours long trip in subspace.
If humanity were to develop this technology in the year 2050,given current projected population growth figures which estimate a global human population at that point of 9.7 billion. And assuming we had the means to expand,how long would it take for humanity to colonize all systems within 30 Light Years of Earth?
Note these colonies need not be on a planet;but may instead be orbital colonies.
Edit 1:Currently within 30 LY of Earth there are confirmed to be 47 star systems,each averaging 2-3 planets in each. The actual number of star systems is likely closer to 72 due to a number of them being red and brown dwarf stars which are very faint in luminosity. Number of ships is going to be conservatively estimated at 10 for the first 30yrs. Jumping to 20,then 40,then 80,and so on with each following decade as orbital shipyards are more firmly established. Assume shipyards outputting 4.2 ships every 5 years. And there are 3 shipyards to start with 18 docking points.
Edit 2:Available resources starting out could be stretched to allow as many as 25 ships in the first 30yrs. However this would require a great deal of manpower being moved into Earth orbit and the shipyards being worked constantly for those 30yrs nonstop. Putting a constant drain on resources that could be reallocated to other uses. Assume one FTL capable ship is able to carry 4 shuttles. Each shuttle can safely land OR be used to help set-up an orbital colony. They each carry around 48 workers and equipment optimally. But a purely human and resupply load would be closer to 130 people per shuttle
Edit 3:Assume each colony establishes a new shipyard in system able to build new ships within 5 years of the colony being established.
To send 1 kg of mass out via FTL costs only 83 cents to start. Costing 54 cents after 25yrs when the system becomes more streamlined. Assume United States dollar values for now.