To generate a recursive sequence that represents the given sequence defined by the explicit formula [tex]\( a_n = 1 + 8(n - 1) \)[/tex], we need to follow these steps:
1. Determine the First Term (Initial Condition):
The explicit formula provides a way to calculate any term [tex]\( a_n \)[/tex] based on its position [tex]\( n \)[/tex]. To find the first term, we substitute [tex]\( n = 1 \)[/tex] into the explicit formula.
[tex]\[
a_1 = 1 + 8(1 - 1) = 1 + 8 \cdot 0 = 1
\][/tex]
So, [tex]\( a_1 = 1 \)[/tex].
2. Establish the Recursive Formula:
The sequence increases by a constant amount as we move from one term to the next. We observe how the explicit formula changes when [tex]\( n \)[/tex] changes:
[tex]\[
a_{n+1} = 1 + 8((n+1) - 1) = 1 + 8n
\][/tex]
Compare this with the original [tex]\( a_n = 1 + 8(n - 1) = 1 + 8n - 8 = 8n - 7 \)[/tex].
Therefore, the difference between consecutive terms [tex]\( a_{n+1} \)[/tex] and [tex]\( a_n \)[/tex] is constant:
[tex]\[
a_{n+1} = a_n + 8
\][/tex]
3. Write the Recursive Definition:
We now arrange our findings into a formal recursive formula:
[tex]\[
\begin{array}{l}
a_1 = 1 \\
a_n = a_{n-1} + 8 \quad \text{for } n > 1
\end{array}
\][/tex]
In conclusion, the recursive sequence representing the given explicit formula is:
[tex]\[
\begin{array}{l}
a_1 = 1 \\
a_n = a_{n-1} + 8 \quad \text{for } n > 1
\end{array}
\][/tex]
