There are a couple off issues here. First, is using a 3-pound weight giving you enough of a workout? In other words, when you get to the end of a set of a particular exercise (like bicep curls), does it take everything you have to do those last few reps, maybe even to the point you can't do the last few? That's what you should be aiming for. If not, you need a heavier weight. Also, keep in mind that you may need different weights for different exercises. For example, when I started strength training I used 5lb weights for bicep curls (a larger, stronger set of muscles) and 3lb weights for tricep extensions (a smaller, weaker set of muscles).
If you are working to failure, then absolutely you need one day of rest between. The whole point of strength training is to tear down the muscle fiber and force it to rebuild. When it rebuilds, it comes back a teeny tiny bit stronger, and new muscle fiber can even be added. If you aren't working hard enough to cause that failure and that fiber breakdown, you're not doing yourself any good. If you are working that hard, then your body needs a day of rest to rebuild the fiber. Skip the rest and you just get breakdown on top of breakdown, and no progress.
Studies have shown that if you are strength training the right way, working each muscle twice per week is enough for most people. The gains made by adding a third workout per week are minimal, and may not be worth the effort/time. I train upper body and lower body alternately, Monday through Thursday. So, I train 4 days in a row, but different muscle groups on subsequent days.
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