To find the recursive formula for the sequence $16, 8, 4, \dots$, let’s analyze the pattern in the sequence:

Observing the Pattern

• The sequence starts at 16, then each subsequent term halves the previous term:
  • $16, 8, 4, \dots$
  • This means each term is multiplied by $\frac{1}{2}$ to get the next term.

Recursive Formula

1. The first term $a_1$ is: $a_1 = 16$
2. The recursive relationship for $a_n$ in terms of the previous term $a_{n-1}$ is: $a_n = \frac{1}{2} \cdot a_{n-1}$

Solution

So, the recursive formula can be written as:

• $a_1 = 16$
• $a_n = \frac{1}{2} a_{n-1}$ for $n > 1$

Would you like further explanation on this process or have any questions?

Here are 5 related questions you might find useful:

1. What is a general formula (non-recursive) for the terms of this sequence?
2. How would the recursive formula change if the sequence was $16, 4, 1, \dots$?
3. Can you create a recursive formula if the sequence were an arithmetic sequence instead?
4. How can you determine if a sequence is geometric or arithmetic?
5. What happens to the recursive formula if the common ratio changes?

Tip: Recursive formulas are particularly useful for sequences with a regular pattern, like geometric or arithmetic progressions.