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

Explanation

To find the 20th term of the arithmetic progression (A.P.) where the 5th term is 5 and the 13th term is -3, we can follow these steps:

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

### Step 2: Set up equations for the given terms
From the problem, we know:
- The 5th term \( a_5 = 5 \)
- The 13th term \( a_{13} = -3 \)

Using the formula:
1. For the 5th term:
\[ a + (5 - 1)d = 5 \]
\[ a + 4d = 5 \] (Equation 1)

2. For the 13th term:
\[ a + (13 - 1)d = -3 \]
\[ a + 12d = -3 \] (Equation 2)

### Step 3: Solve the equations
Now we have two equations:
1. \( a + 4d = 5 \) (Equation 1)
2. \( a + 12d = -3 \) (Equation 2)

We can subtract Equation 1 from Equation 2 to eliminate \( a \):
\[ (a + 12d) - (a + 4d) = -3 - 5 \]
This simplifies to:
\[ 12d - 4d = -8 \]
\[ 8d = -8 \]
\[ d = -1 \]

### Step 4: Substitute \( d \) back to find \( a \)
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 4(-1) = 5 \]
\[ a - 4 = 5 \]
\[ a = 5 + 4 \]
\[ a = 9 \]

### Step 5: Find the 20th term
Now that we have both \( a \) and \( d \), we can find the 20th term \( a_{20} \):
\[ a_{20} = a + (20 - 1)d \]
\[ a_{20} = 9 + 19(-1) \]
\[ a_{20} = 9 - 19 \]
\[ a_{20} = -10 \]

### Final Answer
The 20th term of the progression is:
\[ \boxed{-10} \]