Let's start by understanding the given recursive rule for the sequence:
[tex]\[
a_n = a_{n-1} + 7 \quad \text{with} \quad a_1 = 17
\][/tex]
We need to find an explicit formula for [tex]\(a_n\)[/tex]. To do this, let's investigate the sequence for the first few terms:
- When [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = 17
\][/tex]
- When [tex]\(n = 2\)[/tex]:
[tex]\[
a_2 = a_1 + 7 = 17 + 7 = 24
\][/tex]
- When [tex]\(n = 3\)[/tex]:
[tex]\[
a_3 = a_2 + 7 = 24 + 7 = 31
\][/tex]
- When [tex]\(n = 4\)[/tex]:
[tex]\[
a_4 = a_3 + 7 = 31 + 7 = 38
\][/tex]
Now, let's observe a pattern and try to derive the explicit formula. Each term in the sequence can be seen as the initial term [tex]\(a_1\)[/tex], plus a multiple of 7:
[tex]\[
a_2 = 17 + 7 \cdot 1
\][/tex]
[tex]\[
a_3 = 17 + 7 \cdot 2
\][/tex]
[tex]\[
a_4 = 17 + 7 \cdot 3
\][/tex]
It appears that for the [tex]\(n\)[/tex]-th term:
[tex]\[
a_n = 17 + 7(n - 1)
\][/tex]
We can simplify this expression to obtain the explicit formula:
[tex]\[
a_n = 17 + 7n - 7
\][/tex]
[tex]\[
a_n = 7n + 10
\][/tex]
Therefore, the correct choice for the explicit rule is:
[tex]\[
a_n = 7n + 10
\][/tex]
[tex]\(\boxed{a_n = 7n + 10}\)[/tex]
