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Let's break down and solve both questions from the image:

### Question 7:
Given the sequence \(32, 39, 46, \ldots\), we are asked to find the sum of the first 100 terms, \(S_{100}\).

1. **Recognize the Sequence Type**: 
   This sequence appears to be **arithmetic** because the difference between consecutive terms is constant.
   \[
   d = 39 - 32 = 7
   \]
   Therefore, the common difference \(d = 7\).

2. **Formula for the Sum of an Arithmetic Sequence**:
   The sum of the first \(n\) terms of an arithmetic sequence is given by:
   \[
   S_n = \frac{n}{2} \times \left(2a + (n - 1) \cdot d\right)
   \]
   where:
   - \(a\) is the first term,
   - \(d\) is the common difference,
   - \(n\) is the number of terms.

3. **Substitute Known Values**:
   - First term \(a = 32\),
   - Common difference \(d = 7\),
   - Number of terms \(n = 100\).

   Substituting these into the formula:
   \[
   S_{100} = \frac{100}{2} \times \left(2 \times 32 + (100 - 1) \times 7\right)
   \]
   \[
   S_{100} = 50 \times \left(64 + 693\right) = 50 \times 757 = 37,850
   \]
   The correct answer for Question 7 is **C. 37,850**.

---

### Question 8:
Given the sequence \(46, 39, 32, \ldots\), we are asked to find the sum of the first 100 terms, \(S_{100}\).

1. **Recognize the Sequence Type**: 
   This is again an **arithmetic sequence** with a constant difference.
   \[
   d = 39 - 46 = -7
   \]
   Therefore, the common difference \(d = -7\).

2. **Sum of the First 100 Terms**:
   Using the same formula for the sum:
   \[
   S_n = \frac{n}{2} \times \left(2a + (n - 1) \cdot d\right)
   \]
   - First term \(a = 46\),
   - Common difference \(d = -7\),
   - Number of terms \(n = 100\).

3. **Substitute the Values**:
   \[
   S_{100} = \frac{100}{2} \times \left(2 \times 46 + (100 - 1) \times (-7)\right)
   \]
   \[
   S_{100} = 50 \times \left(92 + (-693)\right) = 50 \times (-601) = -30,050
   \]
   The correct answer for Question 8 is **B. -30,050**.

---

### Do you have any questions, or would you like more details on any of the steps?

Here are 5 related questions you might find useful:
1. How do you identify if a sequence is arithmetic?
2. Can you explain how to derive the formula for the sum of an arithmetic series?
3. What is the difference between an arithmetic and a geometric sequence?
4. How would you find the 100th term of an arithmetic sequence?
5. Can an arithmetic sequence have a common difference of 0?

**Tip**: When working with arithmetic sequences, always check for a constant difference between consecutive terms to confirm the sequence type.

Given the sequence 32, 39, 46, ..., what is the value of S100? Given the sequence 46, 39, 32, ..., what is the value of S100?

This solution explains how to find the sum of the first 100 terms in two arithmetic sequences. Using the sum formula for an arithmetic sequence, it demonstrates step-by-step calculations for sequences with common differences of 7 and -7.

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