Let's solve both arithmetic sequence problems given in the image:

Problem 15

The sequence provided is $\{5, 3, 1, -1, \dots\}$.

1. Identify the first term ($a_1$): $a_1 = 5$.
2. Find the common difference ($d$): $d = 3 - 5 = -2$
3. Use the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n - 1)d$
4. Substitute values to find $a_{18}$: $a_{18} = 5 + (18 - 1)(-2)$ $a_{18} = 5 + 17(-2)$ $a_{18} = 5 - 34$ $a_{18} = -29$ Answer: $a_{18} = -29$

Problem 16

The sequence provided is $\{-2, 5, 12, 19, \dots\}$.

1. Identify the first term ($a_1$): $a_1 = -2$.
2. Find the common difference ($d$): $d = 5 - (-2) = 7$
3. Use the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n - 1)d$
4. Substitute values to find $a_{34}$: $a_{34} = -2 + (34 - 1)(7)$ $a_{34} = -2 + 33 \times 7$ $a_{34} = -2 + 231$ $a_{34} = 229$ Answer: $a_{34} = 229$

Let me know if you need further clarification or additional details!

Related Questions:

1. How do you find the sum of the first $n$ terms in an arithmetic sequence?
2. What are the conditions under which a sequence is considered arithmetic?
3. How do you derive the formula for the $n$-th term of an arithmetic sequence?
4. Can an arithmetic sequence have a common difference of zero?
5. How do you solve for $n$ in an arithmetic sequence if you know the value of $a_n$?

Tip:

In arithmetic sequences, if you know two terms and their positions, you can always determine the common difference and use it to find other terms.