M.D. If you want to be able to do 100 pushups, don't try to do as many pushups as possible every day. If you try to do as many pushups as possible, you will tear your muscles.
All training is done by stressing and recovering, and interval sets.
So you go harder on one day, feel sore on the next and go easy until your muscles feel fresh again. On one day, do three pushups, rest and repeat sets of three until your muscle start to feel sore or stiff. If your muscles are sore on the next day, take the day off. If they are fresh, do sets of three again until your muscles feel sore. You will find that you will improve the most by doing many sets of three on one day and taking the next day or days off until your muscles feel fresh again.