To determine the explicit formula for the arithmetic sequence given as [tex]\(2, 7, 12, 17, \ldots\)[/tex], we follow these steps:
1. Identify the first term ([tex]\(a_1\)[/tex]) and the common difference ([tex]\(d\)[/tex]):
- The first term ([tex]\(a_1\)[/tex]) of the sequence is [tex]\(2\)[/tex].
- The common difference ([tex]\(d\)[/tex]) can be found by subtracting the first term from the second term:
[tex]\[
d = 7 - 2 = 5
\][/tex]
2. Understand the explicit formula for the nth term of an arithmetic sequence:
- The explicit formula for the nth term ([tex]\(a(n)\)[/tex]) of an arithmetic sequence is given by:
[tex]\[
a(n) = a_1 + (n - 1) \times d
\][/tex]
3. Substitute the values we identified ([tex]\(a_1 = 2\)[/tex] and [tex]\(d = 5\)[/tex]) into the formula:
- Replace [tex]\(a_1\)[/tex] with [tex]\(2\)[/tex] and [tex]\(d\)[/tex] with [tex]\(5\)[/tex]:
[tex]\[
a(n) = 2 + (n - 1) \times 5
\][/tex]
Thus, the explicit formula for the arithmetic sequence [tex]\(2, 7, 12, 17, \ldots\)[/tex] is:
[tex]\[
a(n) = 2 + (n - 1) \times 5
\][/tex]
So, the correct choice is:
[tex]\[
a(n) = 2 + (n - 1) 5
\][/tex]
