Sure, let's analyze the sequences in the provided image to determine if they follow an arithmetic sequence and solve for the common difference $d$. Then, we will find the next three terms for each sequence.

Sequence 1: 14, 25, 36, 47, ...

First, we check if the sequence is arithmetic by calculating the differences between consecutive terms:

• $25 - 14 = 11$
• $36 - 25 = 11$
• $47 - 36 = 11$

The common difference $d$ is 11.

The next three terms: $47 + 11 = 58$ $58 + 11 = 69$ $69 + 11 = 80$

Thus, the sequence continues as: $14, 25, 36, 47, 58, 69, 80$

Sequence 2: -34, -29, -24, -19, ...

Check the differences:

• $-29 - (-34) = 5$
• $-24 - (-29) = 5$
• $-19 - (-24) = 5$

The common difference $d$ is 5.

The next three terms: $-19 + 5 = -14$ $-14 + 5 = -9$ $-9 + 5 = -4$

Thus, the sequence continues as: $-34, -29, -24, -19, -14, -9, -4$

Sequence 3: 27, 17, 7, -3, ...

Check the differences:

• $17 - 27 = -10$
• $7 - 17 = -10$
• $-3 - 7 = -10$

The common difference $d$ is -10.

The next three terms: $-3 - 10 = -13$ $-13 - 10 = -23$ $-23 - 10 = -33$

Thus, the sequence continues as: $27, 17, 7, -3, -13, -23, -33$

Sequence 4: $\frac{1}{2}, \frac{1}{5}, \frac{1}{8}, \frac{1}{11},$

Check the differences:

• $\frac{1}{5} - \frac{1}{2} = \frac{2 - 5}{10} = -\frac{3}{10}$
• $\frac{1}{8} - \frac{1}{5} = \frac{5 - 8}{40} = -\frac{3}{40}$
• $\frac{1}{11} - \frac{1}{8} = \frac{8 - 11}{88} = -\frac{3}{88}$

Here, the common difference is not consistent, so it is not an arithmetic sequence.

Sequence 5: $x, 2, x + 4$

Check the differences:

• $2 - x$
• $(x + 4) - 2 = x + 2$

To find $x$, we set the differences equal since they must be the same for an arithmetic sequence: $2 - x = x + 2$ $2 - x = x + 2$ $2 = 2x + 2$ $0 = 2x$ $x = 0$

So, the sequence becomes: $0, 2, 4$

The next three terms: $4 + 2 = 6$ $6 + 2 = 8$ $8 + 2 = 10$

Thus, the sequence continues as: $0, 2, 4, 6, 8, 10$

Would you like any further details or have any questions?

Here are 8 related questions to expand on the previous information:

1. Can you identify other types of sequences apart from arithmetic ones?
2. How can you determine if a sequence is geometric instead of arithmetic?
3. What are the formulas for finding the $n$th term of an arithmetic sequence?
4. How would the approach differ for sequences with fractional or decimal common differences?
5. Can you provide an example of a quadratic sequence and its general formula?
6. How do arithmetic sequences relate to real-world applications?
7. What is the sum of the first $n$ terms in an arithmetic sequence?
8. How can arithmetic sequences be used to solve algebraic problems?

Tip: For verifying if a sequence is arithmetic, always check if the difference between consecutive terms remains constant throughout the sequence.