Question

What pattern do you expect for how a current behaves in a series circuit with equal resistances?

Answer 1

For a series of resistances, the current passing through all of them is same. The potential will be divided among them.

the potential is given by

$V = V_1+V_2+----+V_n=i(R_1+R_2+------+R_n)$

Now the equivalent resistance is given by

$iR =i(R_1+R_2+------+R_n)\\ R =R_1+R_2+------+R_n = nR_1$

In the case of series circuits, if one bulb gets disturbed then all the bulbs after it all will be off. The current stop flowing.