The image contains two sequences, and the instruction asks to find the number of terms in each of the arithmetic progressions (APs).

Let's analyze each sequence:

Sequence 1:

$41, 37, 33, \dots, -79$

This is an arithmetic progression where:

• The first term $a_1 = 41$
• The common difference $d = 37 - 41 = -4$
• The last term $l = -79$

The formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n - 1) \cdot d$ Substitute the values: $-79 = 41 + (n - 1) \cdot (-4)$ Solve for $n$: $-79 - 41 = (n - 1) \cdot (-4)$ $-120 = (n - 1) \cdot (-4)$ $n - 1 = \frac{-120}{-4} = 30$ $n = 31$

So, the number of terms in the first sequence is 31.

Sequence 2:

$9, 15, 21, \dots, 219$

This is another arithmetic progression where:

• The first term $a_1 = 9$
• The common difference $d = 15 - 9 = 6$
• The last term $l = 219$

Using the same formula for the $n$-th term: $a_n = a_1 + (n - 1) \cdot d$ Substitute the values: $219 = 9 + (n - 1) \cdot 6$ Solve for $n$: $219 - 9 = (n - 1) \cdot 6$ $210 = (n - 1) \cdot 6$ $n - 1 = \frac{210}{6} = 35$ $n = 36$

So, the number of terms in the second sequence is 36.

Final Answer:

• The first sequence has 31 terms.
• The second sequence has 36 terms.

Would you like more details on any step? Here are 5 related questions:

1. How do you calculate the sum of an arithmetic sequence?
2. What is the significance of the common difference in arithmetic progressions?
3. How can you derive the formula for the $n$-th term of an arithmetic sequence?
4. What are geometric sequences, and how do they differ from arithmetic ones?
5. Can arithmetic progressions be applied to real-world problems?

Tip: In any arithmetic sequence, the common difference determines how the sequence increases or decreases consistently.