When the number of branches in a parallel circuit increases, what happens to the current and total resistance?

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In a parallel circuit, adding more branches gives more paths for current to flow while the supply voltage across each path stays the same. This makes the overall or equivalent resistance smaller, because the conductances add together when paths are in parallel. With a fixed voltage from the source, Ohm’s law says the total current drawn is I = V / R_total, so as the total resistance decreases, the total current increases. The current in each individual branch depends on that branch’s resistance, but the bigger effect of adding more branches is a lower total resistance and a higher total current from the source. If the branches have equal resistance, the total resistance drops roughly in proportion to the number of branches, and the total current rises accordingly.