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.