Infinite Radial Resistors

It is clear from the diagram that,
All the resistors that have the same color, are in Parallel connection with each other (as the terminals are shorted out).

Hence, We can combine each same colored group into a single resistor.

For the first group,
$\frac{1}{R_1} = \frac{1}{1}+\frac{1}{1}+\frac{1}{1}+\ldots$ 8 times $= 8$
$\Rightarrow R_1 = \frac{1}{8}$

Similarly, $R_2 = \frac{1}{16}$ , $R_3 = \frac{1}{32}, \ldots$

Now,
Clearly,
All of these resistors are in Series connection with one another.

Thus,
Final Resistance $(R)$ = $R_1+R_2+R_3+\ldots$
$\Rightarrow R = \frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots$
$\Rightarrow R = \frac{\frac{1}{8}}{1-\frac{1}{2}}$

Hence,
$R = k = \frac{1}{4}$
$\Rightarrow 100k = \boxed{25}$

felt so good when I got this hehe Ok so if you treat the branches of resistors in the first circle as being in parallel (the current has to branch out through each resistor) then you can use the resistors-in-parallel rule of adding them up, and you find that this comes to 1/8 ohms.

if the first circle has a resistance of 1/8,the next circle will have 1/16 ohms and 1/32 ohms for the next circle and so on. Because once the current has gone through a circle of resistors, there is no branching out required to get through the next circle, you can treat the effective resistance of each circle as being in series with the other circles, so you can just add up the resistances.

this forms a geometric series, 1/8 + 1/16 + 1/32 .... =1/8 (1 + 1/2 + 1/4 + ...) =1/4=k

the question asks for 100k, which equals 25.

We draw an equivalent circuit for the above problem. Then, it can be easily seen that the net resistance between two radial connecting wires is the parallel combination of all resistances between those wires. All radial wires can be assumed to be connected in series with the parallel combinations connecting them. Hence, resistance between center and 1st radial wire is 1/8 ohm, between 1st and 2nd is 1/16 ohm and so on. Adding them in series we get 1/8+1/16+1/32+...... which is equal to (1/8)/(1-1/2)=1/4=0.25 ohm.

since all the circles are conducting wires they bring all the resistors connected to that point at same potential. therefore we can assume them a parallel combination of resistors with 8 connected parallel in first and then 16 connected parallel to each other but in series with equivalent resistance of first 8 resistor which comes out to be 1/8 ohm. thus we get a infinite G. P. like 1/8 +1/16 +1/32 +..... with first term 1/8 and common ratio 1/2 so sum comes out to be (1/8)/(1-1/2)= 0.25 ohm and hence the answer becomes $\boxed{25}$

Resistors between 2 consecutive rings are in parallel (the ringed wire implies that the all points have the same potential) The resulting resistance of each ring is then in series. You end up with a summation of geometric series to infinity with common ratio of half, with the 1st term starting from 1/8. Thus, R = (1/8)*(2) = (1/4)