To find the twentieth term of the arithmetic progression (AP), we need to first determine the common difference of the AP.
Given:
- The first term \( a = 3 \)
- The 12th term, denoted as \( a_{12} = -41 \)
The formula for the \( n \)-th term of an arithmetic progression is given by:
\[ a_n = a + (n-1)d \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.
For the 12th term:
\[ a_{12} = a + (12-1)d = a + 11d \]
Plugging in the known values:
\[ -41 = 3 + 11d \]
Now, we solve for \( d \):
\[ -41 - 3 = 11d \]
\[ -44 = 11d \]
\[ d = \frac{-44}{11} = -4 \]
Now that we have the common difference \( d \), we can find the 20th term \( a_{20} \) using the formula for the \( n \)-th term:
\[ a_{20} = a + (20-1)d = a + 19d \]
Substituting \( a \) and \( d \):
\[ a_{20} = 3 + 19(-4) \]
\[ = 3 - 76 \]
\[ = -73 \]
Therefore, the 20th term of the arithmetic progression is:
\[ \boxed{-73} \]