Growth estimation formula

I was thinking yesterday of how cool it would be to open an investment account for your kids when they're born and deposit $2000 a year in each one. To figure out how much money they would have after n years on a calculator isn't easy, due to the addition of money each year. In a flash of inspiration, though, I realized that the total growth could be calculated easily if you pad the initial deposit such that the interest earned in the first year equals the amount you would normally deposit. Then at the end, subtract the initial padded amount.

For instance, how much money would your child have at age 20? Well, assuming 10% return, you would need $20k to generate your $2k payment. So we use the simple compound interest formula to get 20000 * 1.1^20 = 134550. Subtract the original 20000 and you're left with 114550. How does this compare to using, say, bankrate.com? They come up with 125052, which is pretty close. How cool of a college graduation present would that be?

Is there a more accurate estimation that's still easy to compute? I play these number fantasy games all the time, so any tips would be appreciated. :)