**Derivation of Equivalent Resistance in Series Combination:**

When resistors are connected in series, the current flowing through each resistor is the same, and the total potential difference across the combination is equal to the sum of the potential differences across each resistor.

Let's consider a series combination of 'n' resistors (R1, R2, R3, ..., Rn). The potential difference across the combination is V and the total current flowing through the combination is I.

According to Ohm's Law, the potential difference across each resistor is given by:

V1 = IR1
V2 = IR2
V3 = IR3
...
Vn = IRn

The total potential difference across the combination is given by:

V = V1 + V2 + V3 + ... + Vn
V = I(R1 + R2 + R3 + ... + Rn)

The total resistance (Rs) of the series combination is defined as the ratio of the total potential difference to the total current:

Rs = V/I
Rs = (R1 + R2 + R3 + ... + Rn)

**Derivation of Equivalent Resistance in Parallel Combination:**

When resistors are connected in parallel, the potential difference across each resistor is the same, and the total current flowing through the combination is equal to the sum of the currents flowing through each resistor.

Let's consider a parallel combination of 'n' resistors (R1, R2, R3, ..., Rn). The potential difference across each resistor is V and the total current flowing through the combination is I.

According to Ohm's Law, the current flowing through each resistor is given by:

I1 = V/R1
I2 = V/R2
I3 = V/R3
...
In = V/Rn

The total current flowing through the combination is given by:

I = I1 + I2 + I3 + ... + In
I = V(1/R1 + 1/R2 + 1/R3 + ... + 1/Rn)

The reciprocal of the total resistance (Rp) of the parallel combination is defined as the sum of the reciprocals of the individual resistances:

1/Rp = 1/R1 + 1/R2 + 1/R3 + ... + 1/Rn
Rp = 1/(1/R1 + 1/R2 + 1/R3 + ... + 1/Rn)

**Relationship between Equivalent Resistances:**

The equivalent resistance in a series combination (Rs) is always greater than any individual resistance in the combination. This is because the total resistance is the sum of the individual resistances.

On the other hand, the equivalent resistance in a parallel combination (Rp) is always smaller than any individual resistance in the combination. This is because the total resistance is the reciprocal of the sum of the reciprocals of the individual resistances.

Mathematically, we can express this relationship as:

Rs > R1, R2, R3, ..., Rn
Rp < r1,="" r2,="" r3,="" ...,="" />

Therefore, in a series combination, the equivalent resistance increases as more resistors are added, while in a parallel combination, the equivalent resistance decreases as more resistors are added.

This relationship can be intuitively understood by considering the flow of current. In a series combination, the current has to pass through each resistor successively, leading

In a series combination same current I is flowing through R1,R2,R3.....,Rn. these resistances are connected in series. Here the circuit is connected with the battery of volt V. ==>>The sum of voltages of operating resistances= V==>IR=V==> IR=v1+v2+v3+.....+vn==>R=(v1+v2+....vn)/I==>R=( R1+R2+R3+....Rn)-------eq resistance in series (Rs)... 2) For parallel combination, here we are going to take sum of currents (in each line.)===> R(I1+I2+I3+....In)=V==>(I1+I2+I3+....In)/V=1/R==>(1/R1+1/R2+......1/Rn)=1/R----eq resistance in parallel combination (Rp)