Thanks for you input. I have been online chasing this problem and have tried your solutions, and 2 others suggested by Apple (on the phone) and unfortunately, I've had little success. My iMac is about 10 years old and things are bound to go wrong for any computer being around that long. It seems like the problem, whatever it is, is caused by a software/hardware problem EARLY in the boot process. I am not familiar with this process. I have seen on the internet several people, who seem quite knowledgeable of this process and talk about replacing a certain chip on the motherboard.
I have now begun to study Telepathic Appleography. This is the science, and I use that term lightly, of solving problems of Apple computers my mind (whatever is left of it) by wishing they would just go away.