This is all anybody needs to understand about cycling a tank. This is a classic sigmoid growth curve, applied to bacterial populations here:
Let’s say you have a brand new 300 gal tank. One handful of sand or rubble out of an established tank puts you at the area circled in red. If you add several fish, you will fairly quickly be in the yellow (exponential) phase. The limiting factors of [NH3] and available space are in abundance, so population growth will be exponential.
@Lasse has probably forgotten more about nitrifying bacteria than I know, so I’ll let him chime in on how long it would take to get from early exponential phase (red) to transitional phase (green), but in my experience, it is probably less than a week.
Once at plateau, do whatever you want…tank is fully cycling with bacteria close to carrying capacity (or able to reach K within a day or two as [NH3] increases from fish load)
Where I prefer handful of sand/rubble/live rock from existing system over bottled bacteria:
a) it’s free
b) you are more likely to get the a more biodiverse bacterial community that has already existed in the exact environment you’re adding them to. Does anybody really know exactly what’s in those bottles?