A voltmeter has a resistance `G` and range `V`. Calculate the resistance to be used in series with it to extend its range to `nV`.

Explanation

To solve the problem of calculating the resistance to be used in series with a voltmeter to extend its range from \( V \) to \( nV \), we can follow these steps:

### Step 1: Understand the Configuration
A voltmeter consists of a galvanometer with a resistance \( G \). To extend its range, we need to add a resistance \( R \) in series with the voltmeter.

### Step 2: Define the Current through the Galvanometer
Let the current through the galvanometer when measuring the voltage \( V \) be \( I_G \). This current can be expressed as:
\[ I_G = \frac{V}{G} \]
where \( V \) is the voltage range of the voltmeter and \( G \) is the resistance of the voltmeter.

### Step 3: Write the Expression for the New Voltage Range
When we want to extend the range to \( nV \), the new potential difference across the voltmeter can be expressed as:
\[ V_{AB} = nV \]
Using the same current \( I_G \), we can express \( V_{AB} \) in terms of \( G \) and the new series resistance \( R \):
\[ V_{AB} = I_G (G + R) \]

### Step 4: Substitute \( I_G \) in the New Expression
Substituting \( I_G = \frac{V}{G} \) into the equation gives:
\[ nV = \left(\frac{V}{G}\right)(G + R) \]

### Step 5: Simplify the Equation
Now, we can simplify the equation:
\[ nV = \frac{V(G + R)}{G} \]
Cancelling \( V \) from both sides (assuming \( V \neq 0 \)):
\[ n = \frac{G + R}{G} \]

### Step 6: Rearranging to Find \( R \)
Rearranging the equation to solve for \( R \):
\[ nG = G + R \]
\[ R = nG - G \]
\[ R = G(n - 1) \]

### Final Result
Thus, the resistance \( R \) that needs to be connected in series with the voltmeter to extend its range to \( nV \) is:
\[ R = G(n - 1) \]