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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.

The sequence in the box shows the number of text messages Karis sends each week for four consecutive weeks: 4, 8, 16, 32,... Which formula could be used to determine the number of text messages Karis will send next week (an+1) if the number of text messages sent this week (an) is known?

Learn how to identify the correct formula for a geometric sequence with the example sequence 4, 8, 16, 32. This problem covers key concepts in sequences, including determining the recursive formula and understanding the common ratio. Ideal for students in grades 8-10.

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