My well pump cycles on approx every 3 minutes, even though there aren't any faucets on. The pump is 6 years old and the pressure tank (bladder type) is 18 years old.
You are going to kill that pump with 20 cycles per hour. It sounds like the bladder is gone on the pressure tank. You should have a schrader valve on the the tank. Turn the pump off and then relieve system water pressure. You should have about 2 lbs less pressure in the bladder than the cut-in pressure switch is set for. I'll bet the bladder is gone and you have zero pressure at the valve. If you find you need to buy a new tank, get one large enough to keep the pump cycles to no more than 6 per hour...at least that is the figure I have used for the past umpteen years (never had a pump failure yet...knock on wood). I'm not a well guy (no pun intended ), but I've always lived out in the sticks and had to do my own well pump/pressure systems. Hope this helps some.
If your pump comes on every 3 minutes even though no faucets are on, you have a leak somewhere. The leak could be anywhere. Even the check valve could be letting the water leak back down the well when the pump is off. If your pump only runs a few seconds to refill the tank, you might have a bad bladder tank as well. If it takes a minute or two to refill the tank, then the bladder tank is still good and you just have a leak somewhere.