Question

Find the 12th term of the arithmetic sequence -2x-5, x-12, 4x-19

Answer 1

To find the 12th term of the arithmetic sequence given by the terms \(-2x-5\), \(x-12\), and \(4x-19\), we first need to confirm that the sequence is arithmetic and find its common difference.

1. Calculate the first and second terms:
- First term \(a_1 = -2x - 5\)
- Second term \(a_2 = x - 12\)
- Third term \(a_3 = 4x - 19\)

2. Find the common difference:
The common difference \(d\) can be determined by finding \(a_2 - a_1\) and \(a_3 - a_2\), and confirming they are equal.

\[
d = a_2 - a_1 = (x - 12) - (-2x - 5)
\]
\[
d = x - 12 + 2x + 5 = 3x - 7
\]

Next, calculate \(a_3 - a_2\):
\[
d = a_3 - a_2 = (4x - 19) - (x - 12)
\]
\[
d = 4x - 19 - x + 12 = 3x - 7
\]

Since both differences are equal, the sequence is indeed arithmetic with a common difference of \(d = 3x - 7\).

3. Find the \(n\)-th term formula:
The \(n\)-th term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n-1)d
\]
In this case:
\[
a_n = (-2x - 5) + (n-1)(3x - 7)
\]

4. Substitute \(n = 12\) to find the 12th term:
\[
a_{12} = (-2x - 5) + (12 - 1)(3x - 7)
\]
\[
a_{12} = -2x - 5 + 11(3x - 7)
\]
\[
= -2x - 5 + 33x - 77
\]
\[
= (33x - 2x) + (-5 - 77)
\]
\[
= 31x - 82
\]

Thus, the 12th term of the arithmetic sequence is:

\[
\boxed{31x - 82}
\]