To determine the recursive formula for the given sequence, let’s analyze the pattern. We are given the sequence:
[tex]\[ -1, -7, -13, -19, \ldots \][/tex]
1. Identifying the Sequence Pattern:
To better understand the pattern in the sequence, we look at the differences between consecutive terms.
[tex]\[
-7 - (-1) = -6
\][/tex]
[tex]\[
-13 - (-7) = -6
\][/tex]
[tex]\[
-19 - (-13) = -6
\][/tex]
The common difference between each consecutive term in the sequence is [tex]\(-6\)[/tex].
2. Setting Up the Recursive Formula:
In a recursive formula, each term of the sequence is represented in terms of the previous term. Here, we denote the terms of the sequence as [tex]\( f(n) \)[/tex].
Given [tex]\( f(1) = -1 \)[/tex] (the first term):
[tex]\[
f(1) = -1
\][/tex]
Since the difference between each term is [tex]\(-6\)[/tex], the next term can be defined as:
[tex]\[
f(n) = f(n-1) - 6
\][/tex]
3. Verifying the Recursive Formula:
Let's verify a few terms to ensure that our recursive formula works correctly.
- For [tex]\( n=2 \)[/tex]:
[tex]\[
f(2) = f(1) - 6 = -1 - 6 = -7
\][/tex]
- For [tex]\( n=3 \)[/tex]:
[tex]\[
f(3) = f(2) - 6 = -7 - 6 = -13
\][/tex]
- For [tex]\( n=4 \)[/tex]:
[tex]\[
f(4) = f(3) - 6 = -13 - 6 = -19
\][/tex]
This pattern matches the terms given in the sequence.
Therefore, the recursive formula to generate the sequence is:
[tex]\[
f(n) = f(n-1) - 6 \quad \text{with} \quad f(1) = -1
\][/tex]
