To describe the given sequence [tex]\(-34, -21, -8, 5, \ldots\)[/tex] with a recursively defined function, we need to determine two main components:
1. The first term of the sequence.
2. The common difference between each term in the sequence.
Here are the steps to complete each part of the function:
1. Identify the first term:
- The first term of the sequence is the value at the first position.
- From the sequence provided ([tex]\(-34, -21, -8, 5, \ldots\)[/tex]), the first term is [tex]\( -34 \)[/tex].
2. Determine the common difference:
- The common difference in a sequence is found by subtracting the previous term from the next term.
- Subtract the first term from the second term in the sequence:
[tex]\[
-21 - (-34) = -21 + 34 = 13
\][/tex]
- The common difference is [tex]\( 13 \)[/tex].
3. Formulate the recursive function:
- The recursive formula is: [tex]\( f(n) = f(n-1) + \text{common difference} \)[/tex].
- In this case, [tex]\( f(n) = f(n-1) + 13 \)[/tex].
Using this information, we can complete the recursively defined function:
[tex]\[
\begin{array}{l}
f(1) = -34 \\
f(n) = f(n-1) + 13 \\
\text{for } n = 2, 3, 4, \ldots
\end{array}
\][/tex]
Summarizing:
[tex]\[
\begin{array}{l}
f(1) = -34 \\
f(n) = f(n-1) + 13 \\
\text{for } n = 2, 3, 4, \ldots
\end{array}
\][/tex]
So, the correct numbers to fill in the statements are:
[tex]\[
\begin{array}{l}
f(1)= -34 \\
f(n)=f(n-1)+ 13 \\
\text{for } n=2,3,4, \ldots \\
\end{array}
\][/tex]
