Question

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

Answer 1

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:

\[
\boxed{-73}
\]