I'll share my strategy. Next time, take this to the help forum though.
It combines several methods, including, eventually, gold coins. There have been several posts about population growth and a number of game updates talking about it over the last few years I have been gone. I believe the information in them is not entirely accurate.
The key stat to grow population naturally over time is average age. The lower the better. If it gets too high, growth will slow and stop. Higher still, it will recede. Your health index has short-term and long-term effects. Over the long term, it is more damaging to have it too high. I find the middle ground to keep a good welfare index and avoid long-term damage to my average age is about 130.
Also, if and when you choose to use gold coins to add population, your new population tends to look like your current demographics. For this reason, to the most value out of your purchase, it is ideal to add population when your average age is at its lowest.
Finally, you can add population with worker transfers. This is free, and involves training extra high-paying professions, especially high tech executives, and trading them for much higher numbers of low level workers. You can make this trade with players, but you can also do this by taking computer countries that you don't intend to keep.
That's about it. There is no one-trick pony way to add and keep population reliability. If you are propping up your population with gold coins, it can be detrimental if not done right. The best way is an all-of-the-above strategy that watches the key stat of average age. Keep it low, and you can continue to grow population naturally even when you reach very high levels of population.