Let `{t_(n)}` is an A.P If `t_(1) = 20 , t_(p) = q , t_(q) = p , ` find the value of m such that sum of the first m terms of the A.P is zero .

Answer

To solve the problem, we need to find the value of \( m \) such that the sum of the first \( m \) terms of the arithmetic progression (A.P.) is zero. Given that the first term \( t_1 = 20 \), the \( p \)-th term \( t_p = q \), and the \( q \)-th term \( t_q = p \), we can follow these steps:

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

### Step 2: Substitute the known values into the formula.
Given \( t_1 = 20 \), we have:
\[ a = 20 \]
For the \( p \)-th term:
\[ t_p = a + (p-1)d = q \quad \text{(1)} \]
For the \( q \)-th term:
\[ t_q = a + (q-1)d = p \quad \text{(2)} \]

### Step 3: Set up the equations from the terms.
From equation (1):
\[ 20 + (p-1)d = q \implies (p-1)d = q - 20 \quad \text{(3)} \]
From equation (2):
\[ 20 + (q-1)d = p \implies (q-1)d = p - 20 \quad \text{(4)} \]

### Step 4: Subtract equation (3) from equation (4).
Subtracting (3) from (4):
\[ (q-1)d - (p-1)d = (p - 20) - (q - 20) \]
This simplifies to:
\[ (q - p)d = p - q \]
Rearranging gives:
\[ (q - p)d = -(q - p) \]
If \( q \neq p \), we can divide both sides by \( q - p \):
\[ d = -1 \]

### Step 5: Find the common difference \( d \).
We have found that:
\[ d = -1 \]

### Step 6: Use the formula for the sum of the first \( m \) terms of the A.P.
The sum \( S_m \) of the first \( m \) terms of an A.P. is given by:
\[ S_m = \frac{m}{2} (2a + (m-1)d) \]
Substituting \( a = 20 \) and \( d = -1 \):
\[ S_m = \frac{m}{2} (2 \cdot 20 + (m-1)(-1)) \]
This simplifies to:
\[ S_m = \frac{m}{2} (40 - m + 1) = \frac{m}{2} (41 - m) \]

### Step 7: Set the sum to zero and solve for \( m \).
Setting \( S_m = 0 \):
\[ \frac{m}{2} (41 - m) = 0 \]
This gives us two cases:
1. \( m = 0 \)
2. \( 41 - m = 0 \) which leads to \( m = 41 \)

### Conclusion
The value of \( m \) such that the sum of the first \( m \) terms of the A.P. is zero is:
\[ \boxed{41} \]