I like to approach such questions by starting with simple approximations, and then get more complex only as needed.
To really asses the voltage drop accurately, you would proceed as you suggested; calculating the voltage drop from the source to the first light with the full current, then calculating the drop to the second light with the slightly lower current, then the third light, etc. But you would then need to include the fact that the current drawn by the lights would change with the voltage; the lamp might draw .833A at 120V, but will draw less than this at 115V. If you really want to get picky, you would need to include the fact that the resistance of the copper wire will change with temperature
On the other hand, if you simply assume a lumped load of all of the lights at the end of the line, and assume that the wire is at its 75C resistance, then you will over-estimate voltage drop. If the voltage drop is low enough in this case, then you will be certain that it is low enough with the more complex distributed load.
I'd use #12 conductors and a multi-wire circuit with the lamps evenly distributed on each side. You have a total load of 4.16A at 240V. Resistance of #12 stranded conductors is 2.05 ohms/kft (Chapter 9, table 8, stranded coated conductors). Voltage drop of 3.6% in this worst case; better in reality. 3.6% is a touch high for a branch circuit, but this is a very stable load, so no flickering.
Although you might want to factor in that 100W every 50feet is not that much, and consider the chance that the owner will want to increase the wattage later on.