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Let's analyze and solve the problems shown in the image.

### Problem 1: Fill in the missing numbers in the arithmetic sequence.

We are given:
- \( n: 1, 2, 3, 4, 5, 6 \)
- \( f(n): 10, \_, \_, \_, \_, 25 \)

#### Step 1: Identify the common difference (\( d \)).
- The first term is \( f(1) = 10 \).
- The last term is \( f(6) = 25 \).

Since this is an arithmetic sequence, the difference between consecutive terms is constant. To find the common difference \( d \), we can use the formula for the \( n \)th term of an arithmetic sequence:

\[
f(n) = f(1) + (n-1) \times d
\]

For \( n = 6 \):

\[
25 = 10 + (6-1) \times d 
\]

Simplifying:

\[
25 = 10 + 5d \implies 15 = 5d \implies d = 3
\]

#### Step 2: Fill in the missing numbers.
Now that we know \( d = 3 \), we can find the missing terms:

- \( f(2) = 10 + 1 \times 3 = 13 \)
- \( f(3) = 10 + 2 \times 3 = 16 \)
- \( f(4) = 10 + 3 \times 3 = 19 \)
- \( f(5) = 10 + 4 \times 3 = 22 \)

So the sequence is \( 10, 13, 16, 19, 22, 25 \).

### Problem 2: Explicit formula for the sequence.

The general formula for the \( n \)th term of an arithmetic sequence is:

\[
f(n) = f(1) + (n-1) \times d
\]

Substituting the known values \( f(1) = 10 \) and \( d = 3 \):

\[
f(n) = 10 + (n-1) \times 3 = 3n + 7
\]

So the explicit equation is \( f(n) = 3n + 7 \).

### Summary:
- **Missing numbers in the sequence:** \( 13, 16, 19, 22 \).
- **Common difference \( d \):** \( 3 \).
- **y-intercept (\( f(1) \)):** \( 10 \).
- **Explicit equation:** \( f(n) = 3n + 7 \).

### Additional Questions:
1. How do you determine the common difference in an arithmetic sequence?
2. What is the formula for the \( n \)th term in an arithmetic sequence?
3. How can you find the y-intercept of an arithmetic sequence?
4. What role does the common difference play in forming the sequence?
5. How can you verify if a sequence is arithmetic?

### Tip:
When dealing with sequences, always check the difference between consecutive terms to determine if it's arithmetic or geometric.

Fill in the missing numbers in the arithmetic sequence, identify the common difference, find the y-intercept, and determine the explicit equation for the sequence.

Solve the arithmetic sequence by finding the missing numbers, identifying the common difference, determining the y-intercept, and deriving the explicit formula. The sequence is presented for Grade 6-8 students, covering key concepts in arithmetic progression.

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