Question

Find the 7th term of the arithmetic sequence -3x+9, 2x+13, 7x+17

Answer 1

To find the 7th term of the arithmetic sequence given by the terms \(-3x + 9\), \(2x + 13\), and \(7x + 17\), we first need to determine the common difference of the sequence.

1. **Calculate the common difference:**
- The first term \(a_1\) is \(-3x + 9\).
- The second term \(a_2\) is \(2x + 13\).
- The common difference \(d\) can be calculated as:
\[
d = a_2 - a_1
\]
- Substituting for \(a_1\) and \(a_2\):
\[
d = (2x + 13) - (-3x + 9)
\]
\[
d = 2x + 13 + 3x - 9
\]
\[
d = 5x + 4
\]

2. **Verify with the next term:**
- Now, we check by calculating \(a_3 - a_2\):
- The third term \(a_3\) is \(7x + 17\). Calculate:
\[
a_3 - a_2 = (7x + 17) - (2x + 13)
\]
\[
= 7x + 17 - 2x - 13
\]
\[
= 5x + 4
\]
- Since both differences are equal, the common difference \(d = 5x + 4\) is confirmed.

3. **Finding the 7th term:**
- The nth term of an arithmetic sequence can be found using the formula:
\[
a_n = a_1 + (n - 1)d
\]
- Substituting \(n = 7\):
\[
a_7 = a_1 + 6d
\]
- We already found \(a_1 = -3x + 9\) and \(d = 5x + 4\):
\[
a_7 = (-3x + 9) + 6(5x + 4)
\]
- Calculating this, we have:
\[
a_7 = -3x + 9 + 30x + 24
\]
\[
= (-3x + 30x) + (9 + 24)
\]
\[
= 27x + 33
\]

Thus, the 7th term of the arithmetic sequence is:

\[
\boxed{27x + 33}
\]