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What does it mean if I can perform 25% more reps on my 2nd working set of tricep exercises than my 1st working set of the same weight? And does it mean I'm failing my warmup sets or something else?
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This is a context where what works for different people tends to vary a lot, so experiment with a few different strategies and find what you like best! As long as you're pushing yourself close to muscular failure for a couple sets, you probably will see an increase in strength regardless of the particular loading strategy.
Usually it's best for warmup sets to not be fatiguing at all - you want to save your energy for the heavy working sets close to failure that will provide the most stimulus. So I'd go for 6-8 reps at about half the weight you want to do in your first working set. Warmups are just to get comfortable with the movement and make any adjustments before you do your working sets.
You may like "wave loading" - progressively lowering the reps and increasing the weight over the course of your sets. All sets are performed within a rep or two of failure, but e.g. the first set is failure after 8-10 reps, second set is failure after 6-8 reps, third is failure after 4-6 reps, or something similar. You can use a 1RM calculator to estimate what difference in weight would bring you down from one rep range to another.
It may or may not also be helpful to bring yourself completely to failure in the first set to ensure you're stopping only when you cannot possibly do more reps, and not due to sensation of weakness, slowing of the movement, or moderate reduction in range. To do this, you'll want to keep doing reps until the cable absolutely will not move - you might have a few more reps in the tank on your first set than you realize. Raising the rep range a bit can also help with this.
(also, re: another question from the original post, you can do less than 6-7 reps of a movement and get stronger if you're getting close to muscular failure. I do a lot of sets in the 3-5 rep range, not all the time, but especially for big compound movements it can be great.)