To find the explicit formula for the sequence [tex]\(-19, -28, -37, -46, \ldots\)[/tex], we need to identify the common difference and the first term, then use these to form the general formula for the [tex]\(n\)[/tex]-th term of the arithmetic sequence.
### Step-by-Step Solution:
1. Identify the common difference, [tex]\(d\)[/tex]:
- Find the difference between consecutive terms in the sequence.
- Second term: [tex]\(-28\)[/tex], First term: [tex]\(-19\)[/tex]
[tex]\[
d = -28 - (-19) = -28 + 19 = -9
\][/tex]
- Third term: [tex]\(-37\)[/tex], Second term: [tex]\(-28\)[/tex]
[tex]\[
d = -37 - (-28) = -37 + 28 = -9
\][/tex]
- Fourth term: [tex]\(-46\)[/tex], Third term: [tex]\(-37\)[/tex]
[tex]\[
d = -46 - (-37) = -46 + 37 = -9
\][/tex]
- Hence, the common difference is [tex]\(d = -9\)[/tex].
2. Identify the first term, [tex]\(a_1\)[/tex]:
- The first term of the sequence is given as [tex]\(-19\)[/tex].
3. Formulate the general term, [tex]\(a_n\)[/tex]:
- The general formula for an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
- Substituting [tex]\(a_1 = -19\)[/tex] and [tex]\(d = -9\)[/tex]:
[tex]\[
a_n = -19 + (n-1)(-9)
\][/tex]
- Simplify the expression:
[tex]\[
a_n = -19 + (-9n + 9)
\][/tex]
[tex]\[
a_n = -19 - 9n + 9
\][/tex]
[tex]\[
a_n = -10 - 9n
\][/tex]
4. Evaluate the given options:
- [tex]\(a_n = -10-9n\)[/tex]
- [tex]\(a_n = -10+9n\)[/tex]
- [tex]\(a_n = -19-9n\)[/tex]
- [tex]\(a_n = -19+9n\)[/tex]
Comparing our simplified general formula [tex]\(-10 - 9n\)[/tex] with the given options, we can see that the exact match is:
[tex]\[
a_n = -19 - 9n
\][/tex]
Thus, the correct explicit formula for the given sequence is:
[tex]\[
\boxed{a_n = -19 - 9n}
\][/tex]
