If you have a standard bladder tank on your well, the pump should not cut on and off that rapidly under any circumstance. That makes me think that your bladder tank is bad.
It is not unknown for a reverse osmosis system to malfunction and constantly backwash to the drain. This could cause the pump to cycle on and off, but like I mentioned, a bladder tank system should not cycle nearly as quickly as you are describing.
Lake County, Florida plumber
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