How Does Circuit Resistance Change with Increasing Triangle Numbers?

[Petrulis]

Homework Statement

First of all, there is an an attachment called EXPLANATION which shows the situation and what I have done so far.

So we have got some circuits which are made of triangles (each triangle is made of three resistors (resistors here are shown as rectangles)). The resistance of each resistor is R. The shown part of the circuit is connected to the source with points A and B.

We increse the number of triangles. The first circuit is made of n = 1 triangle, the second circuit has got n = 3 triangles and so on (so n can be equal to 1;3;6;10;15;21...).

I need to find out how the resistance of the circuit depends on n. So I really have to find out the function
(resistance of the circuit) = f(n).

So as you see I can calculate the resistance when n =1;3 and 6, but I stop when I reach n=10.

Homework Equations

Each circuit has got vertical symmetry axis (it is the grey line in the EXPLANATION drawing. This grey line is not the part of the circuit). So the points a, b, c, d, e has got the same potentials, and when we draw the equivalent circuit, we can connect these points with a wire.

The Attempt at a Solution

So I can't draw the equivalent circuit when n = 10, and I can't find out how the resistance of the circuit depends on n.
Maybe I should use other techniques for simplifying the circuit?

Thanks in advance.

 

[Petrulis]

Pleas someone have a look at this problem. It is very important for me.

 

[ogb p]

I would suggest using Kirchhoff's laws and Ohm's law to analyze the circuit and determine the resistance. Kirchhoff's laws state that the sum of currents entering a node must equal the sum of currents leaving the node, and the sum of voltage drops in a closed loop must equal the sum of voltage sources. Ohm's law states that the current through a resistor is directly proportional to the voltage across it and inversely proportional to its resistance.

To simplify the circuit, you can use series and parallel combinations of resistors. In series, the resistors add up to give a total resistance, while in parallel, the inverse of the total resistance is equal to the sum of the inverses of each individual resistance. By using these combinations, you can reduce the circuit to a simpler form and calculate the resistance.

Another approach could be to use the concept of equivalent resistance. This is where you replace a complicated circuit with a single resistor that has the same resistance as the original circuit. The equivalent resistance can be calculated using a formula, and it would make it easier to determine the dependence of resistance on n.

Overall, my suggestion would be to use a combination of these techniques to simplify the circuit and calculate the resistance as a function of n. This will help you understand the relationship between resistance and the number of triangles in the circuit.