To find the recursive formula for the geometric sequence $56, -28, 14, -7, \dots$, we need to identify:

1. The first term $d(1)$.
2. The common ratio $r$, which can be found by dividing any term by the previous term.

Step 1: First Term ($d(1)$)

The first term of the sequence is given as:

$d(1) = 56$

Step 2: Common Ratio ($r$)

To find the common ratio, divide the second term by the first term:

$r = \frac{-28}{56} = -\frac{1}{2}$

Thus, the common ratio $r = -\frac{1}{2}$.

Step 3: Recursive Formula

The recursive formula for a geometric sequence is given by:

$d(n) = d(n-1) \cdot r$

Substituting $r = -\frac{1}{2}$, we get:

$d(n) = d(n-1) \cdot \left( -\frac{1}{2} \right)$

Final Answer:

The recursive formula is:

$d(1) = 56$

$d(n) = d(n-1) \cdot \left( -\frac{1}{2} \right)$

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Would you like further explanation on this? Or do you have any questions?

Here are 5 related questions:

1. How do we find the nth term of a geometric sequence?
2. Can you calculate the 5th term of the sequence using the formula?
3. What happens if the common ratio is greater than 1?
4. How can you determine if a sequence is geometric?
5. What is the sum of the first $n$ terms of a geometric sequence?

Tip: Always check the common ratio by dividing consecutive terms to confirm if the sequence is geometric!