Suppose mth term of an A.P. is 1/n and nth term of the A.P. is 1/m. If rth term of the A.P. is 1, then r is equal to

Options

A. mn
B. m+n
C. m-n
D. m+n-1

Answer

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

### Step-by-Step Solution:

1. **Understanding the Terms of A.P.**:
- The m-th term of an A.P. is given by the formula:
\[ a_m = a + (m-1)d \]
where \( a \) is the first term and \( d \) is the common difference.
- According to the problem, we have:
\[ a_m = \frac{1}{n} \]
Therefore, we can write:
\[ a + (m-1)d = \frac{1}{n} \quad \text{(Equation 1)} \]

2. **Using the n-th Term**:
- Similarly, the n-th term of the A.P. is given by:
\[ a_n = a + (n-1)d \]
According to the problem, we have:
\[ a_n = \frac{1}{m} \]
Thus, we can write:
\[ a + (n-1)d = \frac{1}{m} \quad \text{(Equation 2)} \]

3. **Subtracting the Two Equations**:
- Now, we will subtract Equation 2 from Equation 1:
\[ (a + (m-1)d) - (a + (n-1)d) = \frac{1}{n} - \frac{1}{m} \]
- This simplifies to:
\[ (m-1)d - (n-1)d = \frac{1}{n} - \frac{1}{m} \]
- Which can be further simplified to:
\[ (m-n)d = \frac{1}{n} - \frac{1}{m} \]

4. **Finding the Common Difference \( d \)**:
- The right-hand side can be simplified:
\[ \frac{1}{n} - \frac{1}{m} = \frac{m-n}{mn} \]
- Therefore, we have:
\[ (m-n)d = \frac{m-n}{mn} \]
- Assuming \( m \neq n \), we can divide both sides by \( m-n \):
\[ d = \frac{1}{mn} \]

5. **Finding the First Term \( a \)**:
- Now we will substitute \( d \) back into Equation 1 to find \( a \):
\[ a + (m-1)\left(\frac{1}{mn}\right) = \frac{1}{n} \]
- Rearranging gives:
\[ a + \frac{m-1}{mn} = \frac{1}{n} \]
- Subtract \( \frac{m-1}{mn} \) from both sides:
\[ a = \frac{1}{n} - \frac{m-1}{mn} \]
- Finding a common denominator:
\[ a = \frac{m - (m-1)}{mn} = \frac{1}{mn} \]

6. **Finding the r-th Term**:
- The r-th term of the A.P. is given by:
\[ a_r = a + (r-1)d \]
- Substituting \( a \) and \( d \):
\[ a_r = \frac{1}{mn} + (r-1)\left(\frac{1}{mn}\right) \]
- This simplifies to:
\[ a_r = \frac{1 + (r-1)}{mn} = \frac{r}{mn} \]
- According to the problem, this is equal to 1:
\[ \frac{r}{mn} = 1 \]
- Therefore, multiplying both sides by \( mn \):
\[ r = mn \]

### Conclusion:
The value of \( r \) is \( mn \).