To find the explicit formula for the given sequence [tex]\(-6, -2, 2, 6, 10, \ldots\)[/tex], we need to identify the pattern in the sequence. This sequence appears to be an arithmetic sequence, as the difference between consecutive terms is constant. Let's determine this common difference and then find the formula step-by-step.
1. Identify the First Term ([tex]\(a_1\)[/tex]): The first term of the sequence is [tex]\(-6\)[/tex].
2. Calculate the Common Difference ([tex]\(d\)[/tex]):
To find the common difference, subtract the first term from the second term:
[tex]\[
d = -2 - (-6) = -2 + 6 = 4
\][/tex]
3. Write the General Formula for the [tex]\(n\)[/tex]-th Term:
The explicit formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
Substituting the values of [tex]\(a_1\)[/tex] and [tex]\(d\)[/tex] that we found:
[tex]\[
a_n = -6 + (n-1) \cdot 4
\][/tex]
Now, let's compare this formula with the given options:
A. [tex]\(a_n = -6 + (n-1) \cdot 4\)[/tex] \
B. [tex]\(a_n = 14 + (n-1) \cdot 4\)[/tex] \
C. [tex]\(a_n = -6 + (n-1) \cdot (-4)\)[/tex] \
D. [tex]\(a_n = 4 + (n-1) \cdot (-6)\)[/tex]
The correct explicit formula that matches our derived expression is:
[tex]\[
\boxed{a_n = -6 + (n-1) \cdot 4}
\][/tex]
Therefore, the correct answer is option A: [tex]\(a_n = -6 + (n-1) \cdot 4\)[/tex].
