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Let's solve the questions from the image one by one:

### 1. Find the 100th term of the AP 50, 56, 62, ...
Given:
- First term, \( a = 50 \)
- Common difference, \( d = 56 - 50 = 6 \)
  
The formula for the \(n\)-th term of an arithmetic progression (AP) is:
\[
a_n = a + (n-1) \cdot d
\]
For the 100th term (\(n = 100\)):
\[
a_{100} = 50 + (100 - 1) \cdot 6 = 50 + 99 \cdot 6 = 50 + 594 = 644
\]
The 100th term is **644**.

---

### 2. Find the \(n\)-th term of the AP 18, 16.5, 15, ...
Given:
- First term, \( a = 18 \)
- Common difference, \( d = 16.5 - 18 = -1.5 \)

The \(n\)-th term is:
\[
a_n = a + (n-1) \cdot d
\]
\[
a_n = 18 + (n-1)(-1.5)
\]
So, the general formula for the \(n\)-th term is:
\[
a_n = 18 - 1.5(n-1)
\]

---

### 3. Which term of the AP 13, 20, 27, ... is 384?
Given:
- First term, \( a = 13 \)
- Common difference, \( d = 20 - 13 = 7 \)
- \( a_n = 384 \)

Using the formula for the \(n\)-th term:
\[
a_n = a + (n-1) \cdot d
\]
\[
384 = 13 + (n-1) \cdot 7
\]
Solve for \(n\):
\[
384 - 13 = (n-1) \cdot 7
\]
\[
371 = (n-1) \cdot 7
\]
\[
n-1 = \frac{371}{7} = 53
\]
\[
n = 54
\]
So, 384 is the **54th** term.

---

### 4. Which term of the AP 21, 18, 15, ... is 0?
Given:
- First term, \( a = 21 \)
- Common difference, \( d = 18 - 21 = -3 \)
- \( a_n = 0 \)

Using the formula for the \(n\)-th term:
\[
a_n = a + (n-1) \cdot d
\]
\[
0 = 21 + (n-1)(-3)
\]
Solve for \(n\):
\[
0 = 21 - 3(n-1)
\]
\[
3(n-1) = 21
\]
\[
n-1 = \frac{21}{3} = 7
\]
\[
n = 8
\]
So, 0 is the **8th** term.

---

### 5. Can 200 be any term of the AP 3, 7, 11, ...?
Given:
- First term, \( a = 3 \)
- Common difference, \( d = 7 - 3 = 4 \)
- \( a_n = 200 \)

Using the formula for the \(n\)-th term:
\[
a_n = a + (n-1) \cdot d
\]
\[
200 = 3 + (n-1) \cdot 4
\]
Solve for \(n\):
\[
200 - 3 = (n-1) \cdot 4
\]
\[
197 = (n-1) \cdot 4
\]
\[
n-1 = \frac{197}{4} = 49.25
\]
Since \(n-1\) is not an integer, 200 **cannot** be a term in this AP.

---

Would you like details for the remaining problems, or do you have any questions about these solutions? Here are five related questions for further practice:

1. What is the sum of the first 100 terms of the AP 50, 56, 62, ...?
2. How do you find the sum of the first \(n\) terms of an arithmetic progression?
3. What is the general form of the sum formula for an arithmetic series?
4. How would you solve an AP if given the sum instead of the number of terms?
5. Can you derive the formula for the middle term of a finite arithmetic sequence?

**Tip**: Always remember that the difference between consecutive terms is constant in an arithmetic progression, which simplifies finding any term or the sum of terms.

Solve the following arithmetic progression (AP) problems:
1. Find the 100th term of the AP 50, 56, 62, ...
2. Find the nth term of the AP 18, 16.5, 15, ...
3. Which term of the AP 13, 20, 27, ... is 384?
4. Which term of the AP 21, 18, 15, ... is 0?
5. Can 200 be any term of the AP 3, 7, 11, ...?
6. Find the middle term/s of the finite AP 7, 13, 19, ..., 241.
7. Find the number of terms of the finite AP 3, 6, 9, ..., 111.
8. Find the 12th term and the 24th term of the AP 9, 13, 17, ...

Solve arithmetic progression problems, including finding the 100th term, nth term, middle terms, and determining if a number is a term in the series. Detailed step-by-step solutions with key formulas provided. Suitable for Grades 9-10.

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