The sequence shown is $4, 8, 16, 32, \ldots$, which appears to be a geometric sequence where each term is multiplied by 2 to get the next term.

Given this information, let's evaluate each option to find the correct recursive formula:

1. $a_{n+1} = \frac{1}{2} a_n$: This would mean each term is half of the previous term, which does not match our sequence.

2. $a_{n+1} = 2 \times a_n + 4$: This suggests each term is double the previous term plus 4, which also does not fit the pattern.

3. $a_{n+1} = a_n + 4$: This would imply an arithmetic sequence where each term increases by 4, which does not match our geometric sequence.

4. $a_{n+1} = a_n \times 2$: This correctly describes the relationship, as each term is double the previous term.

The correct answer is: $a_{n+1} = a_n \times 2$

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. What is a geometric sequence, and how does it differ from an arithmetic sequence?
2. How can you find the general formula for a geometric sequence?
3. What would happen if the common ratio was a fraction instead of 2?
4. How would you determine the 10th term in this sequence?
5. What are real-world examples of exponential growth similar to this sequence?

Tip: In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio.