Given the sequence -14, -6, -2, 0, 1, ..., find the recursive formula.

To find the recursive formula for the given sequence \(-14, -6, -2, 0, 1, \dots\), we need to determine how each term relates to the previous term.

First, let's look at the differences between consecutive terms:

\[
-6 - (-14) = 8
\]
\[
-2 - (-6) = 4
\]
\[
0 - (-2) = 2
\]
\[
1 - 0 = 1
\]

Notice that the differences between consecutive terms are decreasing: \(8\), \(4\), \(2\), \(1\), \dots.

The first difference itself forms a sequence, which seems to be halving each time.

Thus, the recursive formula can be expressed as follows:

1. Let the sequence be denoted by \(a_n\).
2. Define the difference between consecutive terms as \(d_n = a_{n+1} - a_n\).

Given the observations, we have:
\[
d_1 = 8, \, d_2 = 4, \, d_3 = 2, \, d_4 = 1, \dots
\]
The difference between consecutive terms \(d_n\) follows:
\[
d_{n+1} = \frac{d_n}{2}
\]
Starting with \(d_1 = 8\).

So, the recursive formula for the sequence \(a_n\) can be written as:
\[
a_{n+1} = a_n + d_n \quad \text{where} \quad d_{n+1} = \frac{d_n}{2}, \text{ with } d_1 = 8
\]

Thus, we have a recursive system:
\[
\begin{cases}
a_{n+1} = a_n + d_n \\
d_{n+1} = \frac{d_n}{2}
\end{cases}
\]
with initial conditions:
\[
a_1 = -14, \, d_1 = 8
\]

Would you like further details on how to derive a closed-form expression, or do you have any other questions? 

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Here are some related questions to explore:

1. How would you find the explicit formula for this sequence?
2. What happens to the sequence as \( n \) becomes very large?
3. Can you describe how to solve a similar problem with different initial values?
4. How does the nature of the recursive difference \(d_n\) influence the sequence?
5. What if the differences \(d_n\) decreased by a constant factor other than \(\frac{1}{2}\)?
6. What would happen if the initial term was positive?
7. Can you generalize the pattern of differences to any sequence?
8. What other sequences might have similar recursive relations?

**Tip:** When working with sequences, always check both the initial conditions and how the differences between terms evolve. This often provides insight into the nature of the sequence.

Learn how to find the recursive formula for the sequence -14, -6, -2, 0, 1, ... by analyzing the differences between consecutive terms and establishing a recursive relationship. This problem involves arithmetic sequences and recursive formulas, suitable for high school students.

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