an arithmetic sequence has a second term of 9 and a fourth term of 21. what is the 40th term in the sequence

1 answer

In an arithmetic sequence, each term can be expressed as:

\[ a_n = a_1 + (n-1)d \]

where \( a_n \) is the nth term, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

Given:

  • The second term \( a_2 \) is 9.
  • The fourth term \( a_4 \) is 21.

We can set up the equations for the second and fourth terms using the formula:

  1. For \( a_2 \): \[ a_1 + d = 9 \quad (1) \]

  2. For \( a_4 \): \[ a_1 + 3d = 21 \quad (2) \]

Now, we can solve these equations simultaneously. First, we'll eliminate \( a_1 \) by rearranging equation (1):

From (1): \[ a_1 = 9 - d \]

Now substitute \( a_1 \) into equation (2):

\[ (9 - d) + 3d = 21 \]

Combine like terms:

\[ 9 + 2d = 21 \]

Now, isolate \( d \):

\[ 2d = 21 - 9 \] \[ 2d = 12 \] \[ d = 6 \]

Now that we have \( d \), we can substitute it back into equation (1) to find \( a_1 \):

\[ a_1 + 6 = 9 \] \[ a_1 = 9 - 6 = 3 \]

Now we have \( a_1 = 3 \) and \( d = 6 \).

To find the 40th term \( a_{40} \), we use the formula:

\[ a_{40} = a_1 + (40-1)d \] \[ a_{40} = 3 + 39 \cdot 6 \] \[ a_{40} = 3 + 234 \] \[ a_{40} = 237 \]

Thus, the 40th term in the sequence is:

\[ \boxed{237} \]