The way I described is very popular and works very fast.
Usually when someone has to increase their pushups or pullups very fast–like if that was some particular aim–it would be best to get the most volume in those exercises they can EVERYDAY - NOT going to failure. Like having a chinup bar in an office or at home and just banging some out (FAR AWAY from failure) as often as possible. [/quote]
Interesting, I remember Chad W. explaining that his clients recovery lasted much longer when they went to failure on those types of exercises (like way longer than was justifiable). So for the sake of an extra few reps (which was the “failure” reps), the clients had to wait a few days extra, which he concluded wasn’t worth it.
I have also found this to be the case especially with pullups (when going to failure it’s hard to progress on them). Obviously, I’m talking about as close to proper failure as possible which requires plenty of ramping sets (not just when it feels hard or like I’m near failure).
Like you said, the better idea is to aim for “accumulative fatigue” whereby the fatigue is built up over higher volume as opposed to very high intensity sets to failure. So the OP could do something like this:
Day 1 - 8 sets of 3
Day 5 - 8 sets of 4
Day 9 - 8 sets of 5
From there on you could decide to do high rep sets, or add weight to the waist. Obviously, it probably won’t be that straight forward (you simply stop doing the sets when failure gets close) but you’d aim for that type of max/volume…