It looks like you have provided a sequence of numbers: \( 3, 4, 1, 10, -17, 44, -139 \).

Let's investigate whether there's a specific pattern or rule governing the progression of this sequence, such as:

1. **Arithmetic progression** (common difference between terms).
2. **Geometric progression** (common ratio between terms).
3. **Recursive relation** (relationship between consecutive terms).

### Analysis:
1. Check the **differences** between consecutive terms:
   \[
   4 - 3 = 1
   \]
   \[
   1 - 4 = -3
   \]
   \[
   10 - 1 = 9
   \]
   \[
   -17 - 10 = -27
   \]
   \[
   44 - (-17) = 61
   \]
   \[
   -139 - 44 = -183
   \]
   The differences are: \( 1, -3, 9, -27, 61, -183 \).

2. Check the **second differences** (differences of differences):
   \[
   -3 - 1 = -4
   \]
   \[
   9 - (-3) = 12
   \]
   \[
   -27 - 9 = -36
   \]
   \[
   61 - (-27) = 88
   \]
   \[
   -183 - 61 = -244
   \]
   The second differences are: \( -4, 12, -36, 88, -244 \).

Since the differences are not constant or forming a clear simple progression, this might be a more complex sequence, potentially based on a recursive or polynomial relation.

Would you like me to delve further into a specific method or attempt to find a formula for this sequence?

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Here are 5 related questions to expand on this:

1. What is the general method for identifying patterns in number sequences?
2. How do we find a recursive formula for a given sequence?
3. Could this sequence be part of a larger mathematical context?
4. What tools can be used to analyze sequences automatically?
5. How can generating functions help in solving sequences?

**Tip:** Identifying differences between consecutive terms is a great first step in analyzing number sequences.

Explore how to analyze the sequence 3, 4, 1, 10, -17, 44, -139 by investigating differences between consecutive terms and second differences. Learn about recursive relations and polynomial connections in sequence analysis, suitable for high school students.

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