Let us analyze the given sequence and solve the problem.

The sequence is defined by the formula:

$s_n = 4n + 7, \quad \text{for } n = 0, 1, 2, 3, \dots$

Step 1: Identify the type of sequence

• A geometric sequence has a constant ratio between consecutive terms.
• An arithmetic sequence has a constant difference between consecutive terms.

Let us calculate the first few terms of the sequence by substituting values of $n$:

$s_0 = 4(0) + 7 = 7, \quad s_1 = 4(1) + 7 = 11, \quad s_2 = 4(2) + 7 = 15, \quad s_3 = 4(3) + 7 = 19$

The differences between consecutive terms are:

$s_1 - s_0 = 11 - 7 = 4, \quad s_2 - s_1 = 15 - 11 = 4, \quad s_3 - s_2 = 19 - 15 = 4$

Since the difference is constant, the sequence is arithmetic with a common difference of $4$.

Step 2: Initial Value

The initial value ($s_0$) is $7$.

Conclusion

The sequence is arithmetic with an initial value of $7$ and a common difference of $4$. Thus, the correct answer is:

$\boxed{\text{C: The sequence is arithmetic with an initial value of 7 and a common difference of 4.}}$

Do you want further explanation or more examples? Let me know!

Related Questions:

1. How do you distinguish between arithmetic and geometric sequences?
2. What is the general formula for the $n$-th term of an arithmetic sequence?
3. Can an arithmetic sequence ever have a constant ratio between terms? Why or why not?
4. How would the formula change if the sequence started at $s_0 = 11$?
5. How can you graph the sequence $s_n = 4n + 7$ and interpret it visually?

Tip:

When analyzing sequences, always calculate a few terms and their differences or ratios to determine the type of sequence.