Let's analyze the given sequence and derive the recursive rule step-by-step.
First, let's identify the initial term of the sequence:
- The sequence starts with [tex]\(a_1 = 21\)[/tex].
Next, we need to determine the common difference between consecutive terms in the sequence. Let's look at the terms:
- The first term is [tex]\(21\)[/tex].
- The second term is [tex]\(-9\)[/tex].
To find the common difference, subtract the first term from the second term:
[tex]\[
-9 - 21 = -30
\][/tex]
Now, we can see that each term in the sequence is obtained by subtracting [tex]\(30\)[/tex] from the previous term. Therefore, the common difference is [tex]\(-30\)[/tex].
The recursive formula for an arithmetic sequence can be written as:
[tex]\[
a_n = a_{n-1} + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference.
In this case, the common difference [tex]\(d\)[/tex] is [tex]\(-30\)[/tex]. Substituting [tex]\(d\)[/tex] into the formula, we get:
[tex]\[
a_n = a_{n-1} - 30
\][/tex]
So, the recursive rule for the given sequence is:
[tex]\[
a_n = a_{n-1} - 30
\][/tex]
Summarizing our findings:
- The initial term of the sequence is [tex]\(a_1 = 21\)[/tex].
- The recursive rule for the sequence is [tex]\(a_n = a_{n-1} - 30\)[/tex].
Therefore, the complete recursive description of the sequence is:
[tex]\[
\begin{cases}
a_1 = 21 \\
a_n = a_{n-1} - 30 \quad \text{for } n > 1
\end{cases}
\][/tex]
