Nitrate panic is often overdone in the hobby.. If everything is healthy, don't fret about it!

At the last store I worked at there was a salt display tank with nitrates reading, I kid you not, 800ppm. I could only test it by diluting a sample of the tank water with pure water.
But no, your nitrates won't disappear as quickly, especially in a new tank. I always get a nitrate spike in a new tank as the live rock "recures" and the last little bit of dieoff is processed. If you left the tank alone for a month I'm sure it would go away, but as soon as you start adding any live animals or feeding it won't go down as fast (if at all).
Couple ways you can reduce nitrate:
1) Water changes. Do several large ones in quick succession.
2) Plant life. Beaslbob will surely chime in soon about the benefits of plant life. I suggest doing some searching for refugiums to see what the big deal is.
3) A denitr
ator. Typically you have to build them yourself and I would never bother, but it's an option. Essentially you have a very long channel for the water to move down (like a spool of airline tubing) and once the water reaches the end the oxygen has been depleted from the water and anaerobic, denitrifying bacteria live and work there.
4) A chemical sponge. I put it in the same class as carbon - it's a chemical media you put in a bag and it will adsorb nitrate from the water. I wouldn't bother with this, either. Not at 20ppm, anyways.