Why LEDs Always Need a Series Resistor
Unlike a resistor, an LED's current doesn't scale gradually with voltage — past its forward voltage threshold, current rises extremely steeply for only a small further voltage increase, meaning a directly-connected LED (with no current-limiting element) will draw far more current than it's rated for and burn out almost immediately. A series resistor limits current to a safe level by absorbing the difference between the supply voltage and the LED's forward voltage drop.
The calculation follows directly from Ohm's law: R = (Vin - Vf) / If — supply voltage minus the LED's forward voltage drop, divided by the target current. Multiple LEDs wired in series each add their own forward voltage drop, which must be subtracted from supply voltage collectively before solving for the shared resistor value, while LEDs in parallel each need their own separate resistor, since manufacturing variation means parallel LEDs won't draw identical current if sharing one resistor.
Getting this resistor value wrong in either direction has a real, visible consequence: too high a resistance makes the LED dim; too low burns it out — this isn't a tolerance range you can eyeball reliably.