As you add more bulbs to a series circuit that means that the bulbs are in series to one another, therefore the total resistance is the sum of the individual resistance of the bulbs. If you add bulbs of the same resistance,then the rate at which the resistance changes will increase in a constant manner provided the current source is not altered. For instance if the bulb you are using is rated 20v,60w, then the current passing via the bulbs in series is the square of the voltage divided by the power in this case the current is approximately 7amperes.
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