In an `A.P.`, the `p^(th)` term is `1/q` and the `q^(th)` term is `1/p`. find the `(pq)^(th)` term of the `A.P.`

Options

A. `(p+q)/(pq)`
B. 0
C. `(pq)/(p+q)`
D. 1

Answer

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

### Step 1: Write the formula for the n-th term of an A.P.
The n-th term of an A.P. is given by:
\[ A_n = A + (n - 1)D \]
where \( A \) is the first term and \( D \) is the common difference.

### Step 2: Set up equations for the p-th and q-th terms
According to the problem, we have:
- The p-th term \( A_p = \frac{1}{q} \)
- The q-th term \( A_q = \frac{1}{p} \)

Using the formula for the n-th term, we can write:
\[ A_p = A + (p - 1)D = \frac{1}{q} \quad \text{(1)} \]
\[ A_q = A + (q - 1)D = \frac{1}{p} \quad \text{(2)} \]

### Step 3: Subtract the two equations
Now, we will subtract equation (2) from equation (1):
\[ (A + (p - 1)D) - (A + (q - 1)D) = \frac{1}{q} - \frac{1}{p} \]
This simplifies to:
\[ (p - 1)D - (q - 1)D = \frac{1}{q} - \frac{1}{p} \]
\[ (p - q)D = \frac{1}{q} - \frac{1}{p} \]

### Step 4: Simplify the right-hand side
The right-hand side can be simplified:
\[ \frac{1}{q} - \frac{1}{p} = \frac{p - q}{pq} \]
Thus, we have:
\[ (p - q)D = \frac{p - q}{pq} \]

### Step 5: Solve for D
Assuming \( p \neq q \) (to avoid division by zero), we can divide both sides by \( (p - q) \):
\[ D = \frac{1}{pq} \]

### Step 6: Substitute D back into one of the equations
We can substitute \( D \) back into equation (1) to find \( A \):
\[ A + (p - 1)D = \frac{1}{q} \]
Substituting \( D = \frac{1}{pq} \):
\[ A + (p - 1)\left(\frac{1}{pq}\right) = \frac{1}{q} \]
This simplifies to:
\[ A + \frac{p - 1}{pq} = \frac{1}{q} \]

### Step 7: Solve for A
Now, isolate \( A \):
\[ A = \frac{1}{q} - \frac{p - 1}{pq} \]
Finding a common denominator:
\[ A = \frac{p - 1}{pq} - \frac{p - 1}{pq} = \frac{1}{pq} \]

### Step 8: Find the (pq)-th term
Now we can find the \( (pq) \)-th term:
\[ A_{pq} = A + (pq - 1)D \]
Substituting \( A = \frac{1}{pq} \) and \( D = \frac{1}{pq} \):
\[ A_{pq} = \frac{1}{pq} + (pq - 1)\left(\frac{1}{pq}\right) \]
This simplifies to:
\[ A_{pq} = \frac{1}{pq} + \frac{pq - 1}{pq} = \frac{1 + pq - 1}{pq} = \frac{pq}{pq} = 1 \]

### Final Answer
Thus, the \( (pq) \)-th term of the A.P. is:
\[ \boxed{1} \]