The 5th and 13th terms of an A.P. are 5 and -3 respectively. Find the 20th term of the progression.

Explanation

To solve the problem, we need to find the 20th term of an arithmetic progression (A.P.) given that the 5th term is 5 and the 13th term is -3.

### Step-by-Step Solution:

1. **Identify the terms of the A.P.**:
- The nth 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.

2. **Write equations for the given terms**:
- For the 5th term (\( T_5 \)):
\[ T_5 = A + 4D = 5 \quad \text{(1)} \]
- For the 13th term (\( T_{13} \)):
\[ T_{13} = A + 12D = -3 \quad \text{(2)} \]

3. **Solve the equations**:
- From equation (1):
\[ A + 4D = 5 \implies A = 5 - 4D \quad \text{(3)} \]
- Substitute equation (3) into equation (2):
\[ (5 - 4D) + 12D = -3 \]
Simplifying this:
\[ 5 - 4D + 12D = -3 \]
\[ 5 + 8D = -3 \]
\[ 8D = -3 - 5 \]
\[ 8D = -8 \]
\[ D = -1 \]

4. **Find the value of \( A \)**:
- Substitute \( D = -1 \) back into equation (3):
\[ A = 5 - 4(-1) \]
\[ A = 5 + 4 = 9 \]

5. **Find the 20th term**:
- The 20th term (\( T_{20} \)) can be calculated as:
\[ T_{20} = A + 19D \]
Substituting the values of \( A \) and \( D \):
\[ T_{20} = 9 + 19(-1) \]
\[ T_{20} = 9 - 19 \]
\[ T_{20} = -10 \]

### Final Answer:
The 20th term of the A.P. is \( -10 \).