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I see that the question is about an arithmetic sequence with missing terms. Let's solve this step-by-step:

1. **Part a: State the common difference**
   - An arithmetic sequence follows the rule that the difference between consecutive terms is constant, called the common difference.
   - In the table, we are given \( f(1) = 7 \) and \( f(5) = 99 \). Using these, we can find the common difference.
   - The formula for the \(n\)-th term of an arithmetic sequence is:
     \[
     f(n) = f(1) + (n-1) \cdot d
     \]
     where \(d\) is the common difference.
   - From the table, \( f(5) = 99 \), so:
     \[
     99 = 7 + (5-1) \cdot d
     \]
     \[
     99 = 7 + 4d
     \]
     \[
     99 - 7 = 4d
     \]
     \[
     92 = 4d \quad \Rightarrow \quad d = \frac{92}{4} = 23
     \]
   - So, the common difference is \(d = 23\).

2. **Part b: Determine the missing terms**
   - Now that we know the common difference \(d = 23\), we can fill in the missing terms:
     \[
     f(2) = f(1) + d = 7 + 23 = 30
     \]
     \[
     f(3) = f(2) + d = 30 + 23 = 53
     \]
     \[
     f(4) = f(3) + d = 53 + 23 = 76
     \]
   - So, the missing terms are \( f(2) = 30 \), \( f(3) = 53 \), and \( f(4) = 76 \).

3. **Part c: Write the explicit formula**
   - The explicit formula for the \(n\)-th term in an arithmetic sequence is given by:
     \[
     f(n) = f(1) + (n-1) \cdot d
     \]
     Substituting \( f(1) = 7 \) and \( d = 23 \), we get:
     \[
     f(n) = 7 + (n-1) \cdot 23
     \]
     Simplifying:
     \[
     f(n) = 7 + 23n - 23 = 23n - 16
     \]
   - So, the explicit formula is \( f(n) = 23n - 16 \).

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Do you need more details or have any questions about this?

Here are 5 related questions to expand your understanding:

1. How can you verify if the common difference in an arithmetic sequence is correct?
2. What is the difference between an arithmetic sequence and a geometric sequence?
3. How would the explicit formula change if the first term were different?
4. Can an arithmetic sequence have a common difference of zero? What would that sequence look like?
5. How do you find the sum of the first \(n\) terms in an arithmetic sequence?

**Tip:** Always double-check your common difference by applying it between consecutive terms to ensure consistency in an arithmetic sequence.

The table contains an arithmetic sequence with missing terms. Find the common difference, determine the missing terms, and write the explicit formula.

Learn how to solve arithmetic sequences by finding the common difference, determining missing terms, and writing the explicit formula. This step-by-step guide helps students in grades 8-10 to understand and master arithmetic sequences.

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