The sequence given is:

$100, 50, 25, 12.5, \dots$

Step 1: Identify the Type of Sequence

This sequence is a geometric sequence because each term is obtained by multiplying the previous term by a constant factor.

Step 2: Determine the Common Ratio

To find the common ratio $r$, divide any term by its previous term: $r = \frac{50}{100} = 0.5$

So, the common ratio $r = 0.5$.

Step 3: Write the Recursive Formula

In a geometric sequence, the recursive formula is given by: $a_n = a_{n-1} \cdot r$

For this sequence:

• The first term $a_1 = 100$.
• The common ratio $r = 0.5$.

Thus, the recursive formula is: $a_n = a_{n-1} \cdot 0.5, \quad \text{with } a_1 = 100$

Would you like further details on this problem, or do you have any other questions?

Here are some related questions you might find helpful:

1. How can we find the nth term of a geometric sequence?
2. What is the difference between an arithmetic and a geometric sequence?
3. How do we find the sum of the first n terms in a geometric sequence?
4. Can a geometric sequence have a negative common ratio?
5. What happens to a geometric sequence as $n$ approaches infinity if $|r| < 1$?

Tip: In a geometric sequence, the common ratio can be found by dividing any term by the previous term.