This is not an answer to your question but rather, additional information that may (or may not) be helpful in thinking about this kind of problem.

When I teach introductory circuit classes, I always emphasize the notion of duality which, when mastered, can give you deep insight into many fundamental "rules" of circuit analysis.

The idea is that if you know the answer for, say, a series circuit, you can take the dual of the result and get a correct answer for a seemingly very different problem.

So, here is short list of circuit duals:


  Voltage - Current
  
  Resistance - Conductance
  
  Inductance - Capacitance
  
  Impedance - Admittance
  
  Series - Parallel
  
  Thevenin - Norton


There are others but these will do most of the time.

Ohm's law is usually written as:

\$V = I R \$

To take the dual, replace all the variables in the above equation with their duals:

The dual of Ohm's Law:

\$I = VG \$

where \$G = \dfrac{1}{R} \$

Recall that for resistors in series, resistances add, so that the equivalent resistance is just the sum.

Consider the dual of this, conductances in parallel.

From the principle of duality, parallel conductances add just as series resistances.  So, if you have 3 conductances in parallel, the equivalent conductance is:

\$G_{eq} = G_1 + G_2 + G_3 \$

Now, convert back to resistance:

\$R_{eq} = \dfrac{1}{G_{eq}} = \dfrac{1}{G_1 + G_2 + G_3} = \dfrac{1}{\frac{1}{R_1}+\frac{1}{R_2} +\frac{1}{R_3}}\$

In other words, the equivalent resistance of \$n\$ parallel resistors is the reciprocal of the sum of the reciprocals.

This is the origin of your first formula.

More than 2 resistors in parallel
I need to use an NPN transistor to switch between no current and ground to control an SN74LS08N AND gate. I want a circuit to function as a switch, so that when the switch circuit is being powered, nothing will be outputted as a result, which is how you make the AND gate receive a 1. When it isn't powered, the switch circuit will be connected to ground, which is how you make the AND gate receive a 0.
