Question

Find equivalent resistance between A and B in the following circuit.

Answer 1

Hint: Figure shows resistances connected in series combination between two points. The equivalent resistance in series combination is the sum of all resistances. We use the sum of n terms of arithmetic progression to add all the resistances. The sum of n terms depends on the number of terms, first term and common difference.
Formulas used:
$R={{R}_{1}}+{{R}_{2}}+--+{{R}_{n}}$
$S=\dfrac{n}{2}(2a+(n-1)d)$

Complete answer:
The property of a material by virtue of which it resists the flow of current through it is called resistance. Its SI unit is ohms. A device which resists the flow of current in a circuit is called a resistor.

In the figure shown above, resistors are connected in series between the points A and B. The formula for equivalent resistance is given by-
$R={{R}_{1}}+{{R}_{2}}+--+{{R}_{n}}$
Substituting the value of resistors in the above equation, we get,
$\begin{align}
  & R'=R+2R+3R+4R+--+100R \\
 & \Rightarrow R'=(1+2+3+4+-+100)R \\
\end{align}$
The above equation is a sum of terms of arithmetic progression wherein, $a=1,\,n=100,\,d=1$
Sum of n terms of an AP is given by the following formula-
$S=\dfrac{n}{2}(2a+(n-1)d)$
Here, $S$ is the sum of n terms
$n$ is the number of terms
$a$ is the first term of AP
$d$ is the common difference between two consecutive terms
Given values are substituted in the above equation to get
$\begin{align}
  & S=\dfrac{100}{2}(2\times 1+(100-1)\times 1) \\
 & \Rightarrow S=50\times (2+99) \\
 & \therefore S=5050 \\
\end{align}$
Therefore, the sum of the following terms is $5050$ and substituting the value in the equivalent resistance we get,
$R'=5050R$
Therefore, between points A and B, the equivalent resistance is $5050R$.

Note:
Resistances can also be connected in parallel combination. The inverse of equivalent resistance of parallel combination is the sum of inverse of all resistances. The resistance is the ratio of voltage and current in a circuit by the Ohm’s law. It depends on the dimensions of a material.