Let's analyze the sequences provided in the image:

Sequence 40: -1, ____, ____, ____, 31, 39 (second space)

To find the missing terms in this sequence, we need to identify a pattern.

Step 1: Calculate the differences between the known terms:

• $31 - 39 = -8$

Given the pattern in typical sequences, it's likely the sequence is arithmetic or geometric, but since the difference is not constant, this might involve a different logic. However, without more information, let's assume it's an arithmetic sequence for simplicity.

Step 2: Work backward:

If the sequence has a constant difference:

• Next difference (let's assume) is also $-8$, so $39 - 8 = 31$.
• If we go further back, the difference should remain $-8$, $31 - 8 = 23$.

The pattern might be: $\ldots, 23, 31, 39$

However, since the earlier numbers could differ, more logic might be involved. For simplicity, without additional terms, the most common solution is an arithmetic sequence.

Sequence 41: -1, ____, ____, ____, 31, 39 (third space)

This appears to be identical to Sequence 40 but asks for the third missing space. Following the same pattern:

• The term just before 31 (in Sequence 40) is 23.
• If we assume the pattern applies to the term before that, it might be $15$, resulting in: $-1, \ldots, 15, 23, 31, 39$

Sequence 42: 13, ____, ____, ____, -11, -17 (first space)

Similarly, let's identify the pattern:

• The difference between $-11$ and $-17$ is $-6$.
• The difference before this could be a constant value: $13, \ldots, \ldots, \ldots, -11, -17$

Given these, the third difference should likely be $-6$ or another consistent value. Suppose we try $-6$:

\[ -5 + 6 = 1 \), thus: \[ 1 + 6 = 7 \), finally: \[ 13 - 6 = 7$$ This leads us to suspect that the first term is either $$7$$. ### Sequence 43: `13, ____, ____, ____, -11, -17` (second space) This might follow similar logic as above: \[ 13, 7, 1, -5, -11, -17 \). So, the second term should be `7`. ### Sequence 44: `13, ____, ____, ____, -11, -17` (third space) Following the same logic, the third term should be `1`. ### Final Answers: 1. **Sequence 40 (second space)**: Likely `23` or based on more complex logic (requires assumption about the pattern). 2. **Sequence 41 (third space)**: Likely `15`. 3. **Sequence 42 (first space)**: Likely `7`. 4. **Sequence 43 (second space)**: Likely `7`. 5. **Sequence 44 (third space)**: Likely `1`. Would you like further details on how to confirm these sequences or have any questions? Here are some additional related problems: 1. How can you identify the type of sequence (arithmetic, geometric, etc.)? 2. What is the general formula for the $$n$$th term in an arithmetic sequence? 3. How do you calculate missing terms in a geometric sequence? 4. What are other common types of sequences and their properties? 5. How do you approach solving sequences with mixed or non-linear patterns? **Tip**: Always consider different possible patterns (arithmetic, geometric, quadratic) when dealing with sequences.