If `T_(r)` be the rth term of an A.P. with first term a and common difference d, `T_(m)=1/n` and `T_(n)=1/m` then `a-d` equals

Options

A. (a) `1/(mn)`
B. (b) `1`
C. (c) `0`
D. (d) `1/m+1/n`

Answer

To solve the problem step by step, we will use the properties of an arithmetic progression (A.P.) and the given conditions.

### Step 1: Write the formulas for the terms of the A.P.
The r-th term \( T_r \) of an A.P. can be expressed as:
\[ T_r = a + (r - 1)d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Write the equations for \( T_m \) and \( T_n \)
According to the problem, we have:
\[ T_m = a + (m - 1)d = \frac{1}{n} \quad \text{(1)} \]
\[ T_n = a + (n - 1)d = \frac{1}{m} \quad \text{(2)} \]

### Step 3: Set up the equations
From equations (1) and (2), we can rewrite them as:
1. \( a + (m - 1)d = \frac{1}{n} \)
2. \( a + (n - 1)d = \frac{1}{m} \)

### Step 4: Subtract the two equations
Subtract equation (2) from equation (1):
\[ \left(a + (m - 1)d\right) - \left(a + (n - 1)d\right) = \frac{1}{n} - \frac{1}{m} \]
This simplifies to:
\[ (m - 1)d - (n - 1)d = \frac{1}{n} - \frac{1}{m} \]
\[ (m - n)d = \frac{1}{n} - \frac{1}{m} \]

### Step 5: Simplify the right side
To simplify \( \frac{1}{n} - \frac{1}{m} \):
\[ \frac{1}{n} - \frac{1}{m} = \frac{m - n}{mn} \]
Thus, we have:
\[ (m - n)d = \frac{m - n}{mn} \]

### Step 6: Solve for \( d \)
Assuming \( m \neq n \), we can divide both sides by \( m - n \):
\[ d = \frac{1}{mn} \]

### Step 7: Substitute \( d \) back to find \( a \)
Now we can substitute \( d \) back into either equation (1) or (2) to find \( a \). We'll use equation (1):
\[ a + (m - 1)\left(\frac{1}{mn}\right) = \frac{1}{n} \]
This simplifies to:
\[ a + \frac{m - 1}{mn} = \frac{1}{n} \]
Rearranging gives:
\[ a = \frac{1}{n} - \frac{m - 1}{mn} \]
Finding a common denominator:
\[ a = \frac{m - 1 - (m - 1)}{mn} + \frac{1}{n} = \frac{1}{n} - \frac{m - 1}{mn} = \frac{m - 1 - (m - 1)}{mn} = \frac{1}{mn} \]

### Step 8: Calculate \( a - d \)
Now we can find \( a - d \):
\[ a - d = \frac{1}{mn} - \frac{1}{mn} = 0 \]

### Final Answer
Thus, the value of \( a - d \) is:
\[ \boxed{0} \]