The 12th term of an arithmetic progression is -41. Given that the first term is 3. Find the twentieth term.

1 answer

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:

\[ \boxed{-73} \]