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 $f(n) = 23n - 16$.

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.