The nth term of an A.P. is `(1)/(n)` and nth term is `(1)/(m).` Its (mn)th term is :

Options

A. mn
B. `(1)/(mn)`
C. 1
D. None of these

Explanation

To solve the problem, we need to find the (mn)th term of an arithmetic progression (A.P.) given that the nth term is \( \frac{1}{n} \) and the mth term is \( \frac{1}{m} \).

### Step-by-Step Solution:

1. **Understand the nth term of the A.P.**:
The nth term of an A.P. can be expressed as:
\[ A_n = A + (n - 1)d \]
where \( A \) is the first term and \( d \) is the common difference.

2. **Set up equations for the nth and mth terms**:
From the problem statement, we have:
\[ A_n = \frac{1}{n} \quad \text{(1)} \]
\[ A_m = \frac{1}{m} \quad \text{(2)} \]

Using the formula for the nth term:
- From (1):
\[ A + (n - 1)d = \frac{1}{n} \]
- From (2):
\[ A + (m - 1)d = \frac{1}{m} \]

3. **Subtract the two equations**:
Subtract equation (1) from equation (2):
\[ [A + (m - 1)d] - [A + (n - 1)d] = \frac{1}{m} - \frac{1}{n} \]
This simplifies to:
\[ (m - n)d = \frac{1}{m} - \frac{1}{n} \]

4. **Simplify the right side**:
The right side can be simplified:
\[ \frac{1}{m} - \frac{1}{n} = \frac{n - m}{mn} \]
Thus, we have:
\[ (m - n)d = \frac{n - m}{mn} \]

5. **Solve for d**:
Rearranging gives:
\[ d = \frac{n - m}{mn(m - n)} \]
Since \( m - n \) is negative, we can express \( d \) as:
\[ d = -\frac{1}{mn} \]

6. **Find the first term A**:
Now we can substitute \( d \) back into one of the original equations to find \( A \). Using equation (1):
\[ A + (n - 1)\left(-\frac{1}{mn}\right) = \frac{1}{n} \]
Simplifying gives:
\[ A - \frac{n - 1}{mn} = \frac{1}{n} \]
Rearranging:
\[ A = \frac{1}{n} + \frac{n - 1}{mn} \]
Finding a common denominator:
\[ A = \frac{m + (n - 1)}{mn} = \frac{m + n - 1}{mn} \]

7. **Find the (mn)th term**:
Now, we can find the (mn)th term using the formula:
\[ A_{mn} = A + (mn - 1)d \]
Substituting \( A \) and \( d \):
\[ A_{mn} = \frac{m + n - 1}{mn} + (mn - 1)\left(-\frac{1}{mn}\right) \]
Simplifying gives:
\[ A_{mn} = \frac{m + n - 1}{mn} - \frac{mn - 1}{mn} = \frac{m + n - 1 - (mn - 1)}{mn} \]
Thus:
\[ A_{mn} = \frac{m + n - mn}{mn} \]

### Final Answer:
The (mn)th term of the A.P. is:
\[ A_{mn} = \frac{m + n - mn}{mn} \]