I don't see how you jumped from the +5V above the R_LIMIT to the 3.0V in the R_LIMIT calculation rather than using 5V.

When components are in series their

voltage drops *add* up while their

current *is equal*. --There are only so many electrons.

The circuit in this case is an Arduino Pin, Resistor, and LED.

So if your source (like the Arduino pin) has a voltage of 5V and the LED will drop 2V (its forward voltage), that leaves a 3V drop for the resistor.

Since the parts are in series and you want 20mA to flow through the LED, you use Ohm's Law to calculate the resistance value knowing that WHATEVER the resistor value is, it is going to drop 3V.

Ohm's law says that Resistance is equal to Voltage divided by Current. We know the voltage drop of the resistor (3V) and the current (20mA). That gives 150ohms.