When the rate declines, the growth slows (and you are still ignoring the absolute cap that the land mass puts on population).
You cite the quadrupling of population in the last century. Fine, that is nice, HOWEVER it is not enough. By you own math you require a trippling (3 workers per retiree) of the population EVERY GENERATION (20 to 30 years), not every century. You are only getting about a quarter the growth you need.
Botom line, you do NOT have a long term sustainable system.