I see you've uploaded an image containing sequences and some circled items. Let me extract the mathematical content from this image and provide you with solutions and explanations. I'll analyze it now. PIL import Image import pytesseract

Load the uploaded image

image_path = "/mnt/data/1736797387069xx5qox92.jpg" image = Image.open(image_path)

Use OCR to extract text

extracted_text = pytesseract.image_to_string(image)

Display the extracted text

extracted_textThe OCR has extracted the following content from the image:

11. -18, -13, -9, -6
12. 120, 60, 30, 15
62, 54, 18, 6
15. 2, 5, 11, 20
17. 2, 3, 1, 4, 0
18. 20, 120, 24, 6

It seems to include a mix of arithmetic and geometric sequences. I will analyze the given sequences one by one and solve or describe their patterns. Let's start!### Analysis of the Sequences

11. Sequence: -18, -13, -9, -6

This is an arithmetic sequence, where each term increases by 4.
Rule: $a_n = -18 + (n-1) \cdot 4$
Next Term: $-6 + 4 = -2$.

12. Sequence: 120, 60, 30, 15

This is a geometric sequence, where each term is halved (common ratio = $\frac{1}{2}$).
Rule: $a_n = 120 \cdot \left(\frac{1}{2}\right)^{n-1}$
Next Term: $15 \cdot \frac{1}{2} = 7.5$.

15. Sequence: 2, 5, 11, 20

The differences are:
$5 - 2 = 3$, $11 - 5 = 6$, $20 - 11 = 9$.
The differences increase by 3, suggesting a quadratic pattern.
Rule: $a_n = n^2 + n$
(Verify: $a_1 = 1^2+1=2, a_2=2^2+2=6$, etc.).
Next Term: $4^2 + 4 = 20$.

17. Sequence: 2, 3, 1, 4, 0

The differences between consecutive terms are irregular. Further clarification or a specific rule might be required to predict the next term.

18. Sequence: 20, 120, 24, 6

This sequence alternates between large and small numbers, suggesting it is not straightforwardly arithmetic or geometric. Could you clarify the pattern you're observing or the context?

Would you like detailed solutions for each sequence? Here are some potential follow-up questions:

5 Follow-Up Questions:

1. Would you like me to elaborate on arithmetic and geometric sequences in general?
2. Should I derive the full general term formula for any specific sequence above?
3. Would you like help identifying the pattern in the irregular sequences (e.g., #17)?
4. Is there a specific term you want me to find for any of these sequences?
5. Should I solve similar types of sequences for practice?

Tip: For geometric sequences, multiplying the common ratio repeatedly makes it easy to predict future terms!