Resistors $R, 2R, 4R, 8R, ..... \infty$ are connected in parallel. What is their equivalent resistance?

Three resistances of equal values are arranged in four different configurations as shown below. Power dissipation in the increasing order is

$A$ wire of resistance per unit length $\rho_L = 10^{-6} \, \Omega/m$ is bent into a circle of diameter $2 \, m$. $A$ piece of wire of the same material is connected across the diameter $AB$. Find the resistance between $A$ and $B$.

$3$ identical bulbs are connected in series and these together dissipate a power $P$. If now the bulbs are connected in parallel,then the power dissipated will be

Four resistors, $100 \Omega, 200 \Omega, 300 \Omega$ and $400 \Omega$ are connected to form four sides of a square. The resistors can be connected in any order. What is the maximum possible equivalent resistance across the diagonal of the square (in $\Omega$)?

$A$ ring is made of a wire having a total resistance $R_0 = 12\,\Omega$. Find the ratio of the lengths $\frac{l_1}{l_2}$ of the two segments of the ring between points $A$ and $B$,as shown in the figure,such that the equivalent resistance $R$ of the sub-circuit between these points is equal to $\frac{8}{3}\,\Omega$.