Which of the following is the explicit formula for the sequence: [tex]\(-19, -28, -37, -46, \ldots\)[/tex]?

A. [tex]\(a_n = -10 - 9n\)[/tex]

B. [tex]\(a_n = -10 + 9n\)[/tex]

C. [tex]\(a_n = -19 - 9n\)[/tex]

D. [tex]\(a_n = -19 + 9n\)[/tex]



Answer :

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]