According to the rule of 72, if arielle invests $100, $200, and $2000 into three separate accounts with the same interest rate, which amount will double fastest?

The rule of 72 is an approximate estimate of the time it takes to double an investment, and depends only on the interest rate.  So amount of deposit does not change the estimate.  All three accounts will take the same time to double.

If the accounts are all deposited on the same day with the same interest rate and same compounding period, they all double at the same time, whether using the rule of 72 or the actual time.