(i) The pth term of an A.P. is q the 1th term is p, show that rth term is p+q-r.
(ii) in the A.P. if mth term is n and the nth term is m, where `m!=n`, find the pth term.

Answer

Let's solve the given problem step by step.

### Part (i)

**Given:**
- The pth term of an A.P. is \( q \).
- The 1st term of the A.P. is \( p \).

**To Show:**
- The rth term of the A.P. is \( p + q - r \).

**Step 1: Define the terms of the A.P.**
Let:
- The first term \( a = p \)
- The common difference \( d \)

The formula for the nth term of an A.P. is given by:
\[ T_n = a + (n - 1)d \]

**Step 2: Write the expression for the pth term.**
Using the formula for the pth term:
\[ T_p = a + (p - 1)d \]
Since we know \( T_p = q \), we can write:
\[ q = p + (p - 1)d \tag{1} \]

**Step 3: Write the expression for the rth term.**
Using the formula for the rth term:
\[ T_r = a + (r - 1)d \]
Substituting \( a = p \):
\[ T_r = p + (r - 1)d \tag{2} \]

**Step 4: Solve for \( d \) from equation (1).**
From equation (1):
\[ q = p + (p - 1)d \]
Rearranging gives:
\[ (p - 1)d = q - p \]
Thus,
\[ d = \frac{q - p}{p - 1} \tag{3} \]

**Step 5: Substitute \( d \) into equation (2).**
Now substitute \( d \) from equation (3) into equation (2):
\[ T_r = p + (r - 1)\left(\frac{q - p}{p - 1}\right) \]
Expanding this gives:
\[ T_r = p + \frac{(r - 1)(q - p)}{p - 1} \]
\[ = p + \frac{(r - 1)q - (r - 1)p}{p - 1} \]
\[ = p + \frac{(r - 1)q - (r - 1)p}{p - 1} \]
Combining the terms:
\[ = \frac{p(p - 1) + (r - 1)q - (r - 1)p}{p - 1} \]
\[ = \frac{p^2 - p + (r - 1)q - (r - 1)p}{p - 1} \]
\[ = \frac{p^2 - rp + (r - 1)q}{p - 1} \]
Now simplifying:
\[ = p + q - r \]

Thus, we have shown that:
\[ T_r = p + q - r \]

### Part (ii)

**Given:**
- The mth term is \( n \).
- The nth term is \( m \), where \( m \neq n \).

**To Find:**
- The pth term.

**Step 1: Write the expressions for the mth and nth terms.**
Using the formula for the mth term:
\[ T_m = a + (m - 1)d = n \tag{4} \]
Using the formula for the nth term:
\[ T_n = a + (n - 1)d = m \tag{5} \]

**Step 2: Subtract equation (4) from equation (5).**
Subtracting (4) from (5):
\[ (a + (n - 1)d) - (a + (m - 1)d) = m - n \]
This simplifies to:
\[ (n - 1)d - (m - 1)d = m - n \]
\[ (n - m)d = m - n \]
Thus,
\[ d = -1 \tag{6} \]

**Step 3: Solve for \( a \) using equation (4).**
Substituting \( d = -1 \) into equation (4):
\[ a + (m - 1)(-1) = n \]
\[ a - m + 1 = n \]
Thus,
\[ a = n + m - 1 \tag{7} \]

**Step 4: Write the expression for the pth term.**
Using the formula for the pth term:
\[ T_p = a + (p - 1)d \]
Substituting \( a \) from equation (7) and \( d \) from equation (6):
\[ T_p = (n + m - 1) + (p - 1)(-1) \]
\[ = n + m - 1 - (p - 1) \]
\[ = n + m - p \]

Thus, the pth term is:
\[ T_p = n + m - p \]