To find the common difference of an arithmetic sequence, we need to identify how much each term increases or decreases by as we move from one term to the next.
Consider the given arithmetic sequence: [tex]\(-19, -11, -3, \ldots\)[/tex]
1. Identify the terms:
- The first term ([tex]\(a_1\)[/tex]) is [tex]\(-19\)[/tex].
- The second term ([tex]\(a_2\)[/tex]) is [tex]\(-11\)[/tex].
- The third term ([tex]\(a_3\)[/tex]) is [tex]\(-3\)[/tex].
2. Calculate the common difference:
The common difference ([tex]\(d\)[/tex]) of an arithmetic sequence is found by subtracting the previous term from the current term.
- From the first term to the second term:
[tex]\[
d = a_2 - a_1 = -11 - (-19)
\][/tex]
Simplifying this, we get:
[tex]\[
d = -11 + 19 = 8
\][/tex]
- From the second term to the third term:
[tex]\[
d = a_3 - a_2 = -3 - (-11)
\][/tex]
Simplifying this, we get:
[tex]\[
d = -3 + 11 = 8
\][/tex]
3. Verify consistency:
Both calculations yield the same common difference, confirming that the sequence is indeed arithmetic and that the common difference is consistent between consecutive terms.
So, the common difference of the arithmetic sequence [tex]\(-19, -11, -3, \ldots\)[/tex] is [tex]\(\boxed{8}\)[/tex].
