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:

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$$ Would you like further details on how to derive a closed-form expression, or do you have any other questions? --- Here are some related questions to explore: 1. How would you find the explicit formula for this sequence? 2. What happens to the sequence as $$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.