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### 1st Question: Which of the following is *not* an Arithmetic Progression (AP)?

Let's analyze each sequence:

1. **1, 4, 7, 10,...**  
   - The common difference is \(4 - 1 = 3\), \(7 - 4 = 3\), and \(10 - 7 = 3\).  
   - This is an AP.

2. **3, 7, 12, 16,...**  
   - The common difference is \(7 - 3 = 4\), \(12 - 7 = 5\), and \(16 - 12 = 4\).  
   - The difference is not consistent, so this is *not* an AP.

3. **-5, -2, 1, 4,...**  
   - The common difference is \(-2 - (-5) = 3\), \(1 - (-2) = 3\), and \(4 - 1 = 3\).  
   - This is an AP.

4. **11, 14, 17, 20,...**  
   - The common difference is \(14 - 11 = 3\), \(17 - 14 = 3\), and \(20 - 17 = 3\).  
   - This is an AP.

So, the answer is **3, 7, 12, 16** (Option 2).

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### 2nd Question: Evaluate the geometric series \(3 + 15 + 75 + 375 + \dots\) for \(n = 6\).

This is a geometric progression with:

- First term \(a = 3\)
- Common ratio \(r = \frac{15}{3} = 5\)
- Number of terms \(n = 6\)

The sum \(S_n\) of a geometric series is given by the formula:
\[
S_n = a \frac{r^n - 1}{r - 1}
\]
Substitute the values:
\[
S_6 = 3 \frac{5^6 - 1}{5 - 1} = 3 \frac{15625 - 1}{4} = 3 \frac{15624}{4} = 3 \times 3906 = 11718
\]

The answer is **11,718**.

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Would you like more details on any of these solutions or have any questions? Here are five related questions you could explore:

1. What defines an arithmetic progression?
2. How can you identify a geometric progression from a sequence of numbers?
3. What happens to the sum of a geometric series when the common ratio is less than 1?
4. How do you derive the formula for the sum of an arithmetic progression?
5. Can a sequence be neither arithmetic nor geometric? How would that look?

**Tip:** For geometric series, remember that the common ratio is the key factor; any change in it can significantly alter the sum calculation.

Which of the following is not an Arithmetic Progression? Evaluate the geometric series described: 3 + 15 + 75 + 375 + ..., n = 6.

Learn how to identify a sequence that is not an arithmetic progression and how to evaluate the sum of a geometric series. This problem is suitable for students in Grades 9-12, covering key algebraic concepts.

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