The image contains four problems related to arithmetic or geometric sequences. Let’s break them down one by one.

Problem e:

Find the 21st term of the sequence 18, 14, 10, ...

This is an arithmetic sequence where:

• First term $a_1 = 18$
• Common difference $d = 14 - 18 = -4$

The formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n - 1) \cdot d$ For the 21st term ($n = 21$): $a_{21} = 18 + (21 - 1) \cdot (-4) = 18 + 20 \cdot (-4) = 18 - 80 = -62$

So, the 21st term is -62.

Problem f:

Find the 19th term of the sequence $3\sqrt{2}, 7\sqrt{2}, 11\sqrt{2}, \dots$

This is an arithmetic sequence where:

• First term $a_1 = 3\sqrt{2}$
• Common difference $d = 7\sqrt{2} - 3\sqrt{2} = 4\sqrt{2}$

Using the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n - 1) \cdot d$ For the 19th term ($n = 19$): $a_{19} = 3\sqrt{2} + (19 - 1) \cdot 4\sqrt{2} = 3\sqrt{2} + 18 \cdot 4\sqrt{2} = 3\sqrt{2} + 72\sqrt{2} = 75\sqrt{2}$

So, the 19th term is 75√2.

Problem g:

Find the 30th term of the sequence $\frac{1}{3}, 1, 1\frac{2}{3}, \dots$

This is an arithmetic sequence where:

• First term $a_1 = \frac{1}{3}$
• The common difference $d = 1 - \frac{1}{3} = \frac{2}{3}$

Using the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n - 1) \cdot d$ For the 30th term ($n = 30$): $a_{30} = \frac{1}{3} + (30 - 1) \cdot \frac{2}{3} = \frac{1}{3} + 29 \cdot \frac{2}{3} = \frac{1}{3} + \frac{58}{3} = \frac{59}{3} = 19 \frac{2}{3}$

So, the 30th term is 19⅔.

Problem h:

Find the $n$-th term of the sequence 2, 5, 8, ...

This is an arithmetic sequence where:

• First term $a_1 = 2$
• Common difference $d = 5 - 2 = 3$

Using the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n - 1) \cdot d$ Thus, the formula for the $n$-th term is: $a_n = 2 + (n - 1) \cdot 3 = 2 + 3n - 3 = 3n - 1$

So, the $n$-th term is $3n - 1$.

Let me know if you'd like further clarifications or more details!

Here are five related questions to expand your knowledge:

1. How can you derive the general formula for a geometric sequence?
2. What are the differences between arithmetic and geometric sequences?
3. How do you find the sum of the first $n$ terms in an arithmetic sequence?
4. What changes when the common difference of a sequence is negative?
5. Can an arithmetic sequence have irrational numbers as terms?

Tip: Always check the common difference (or ratio for geometric sequences) to ensure correct identification of the sequence type.