Let's go through the questions one by one:

1. What is the next term in the geometric sequence $324, -108, 36?$

A geometric sequence is defined by a constant ratio between successive terms. To find the common ratio ($r$):

$r = \frac{-108}{324} = -\frac{1}{3}$ Check the second ratio: $r = \frac{36}{-108} = -\frac{1}{3}$ So, the common ratio is $-\frac{1}{3}$.

Now, multiply the last term by $-\frac{1}{3}$ to get the next term: $36 \times -\frac{1}{3} = -12$

Answer: C. -12

2. Find the common difference in the arithmetic sequence $1, \frac{6}{4}, 2, \frac{10}{4}, 3, \dots$

An arithmetic sequence has a common difference between consecutive terms. Let's calculate the difference between each pair of successive terms:

$\frac{6}{4} - 1 = \frac{2}{4} = \frac{1}{2}$ $2 - \frac{6}{4} = \frac{8}{4} - \frac{6}{4} = \frac{2}{4} = \frac{1}{2}$ $\frac{10}{4} - 2 = \frac{10}{4} - \frac{8}{4} = \frac{2}{4} = \frac{1}{2}$ Thus, the common difference is $\frac{1}{2}$.

Answer: C. $\frac{1}{2}$

3. Which set of numbers is an example of a harmonic sequence?

A harmonic sequence is one where the reciprocals of the terms form an arithmetic sequence.

• For option C, the reciprocals are $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}$. This does form an arithmetic sequence because the difference between consecutive reciprocals is constant: $\frac{1}{9} - \frac{1}{3} = -\frac{2}{9}, \quad \frac{1}{27} - \frac{1}{9} = -\frac{2}{27}$

Answer: C.

4. What is the sum of all the even integers between 9 and 27?

The even integers between 9 and 27 are: $10, 12, 14, 16, 18, 20, 22, 24, 26$.

This is an arithmetic sequence with:

• First term $a = 10$,
• Common difference $d = 2$,
• Last term $l = 26$.

The number of terms $n$ is given by: $n = \frac{l - a}{d} + 1 = \frac{26 - 10}{2} + 1 = 9$

The sum $S_n$ of an arithmetic sequence is: $S_n = \frac{n}{2} (a + l) = \frac{9}{2} (10 + 26) = \frac{9}{2} \times 36 = 162$

Answer: B. 162

5. If three arithmetic means are inserted between $-15$ and $9$, find the first of these arithmetic means.

Inserting three arithmetic means between $-15$ and $9$ gives four equal intervals (since the total sequence has 5 terms). The common difference $d$ is:

$d = \frac{9 - (-15)}{4} = \frac{24}{4} = 6$

The first arithmetic mean is: $-15 + 6 = -9$

Answer: D. -9

Let me know if you need further explanations on any of these! Here are some additional questions you might find useful:

1. How do you determine the common ratio in a geometric sequence?
2. What defines a harmonic sequence mathematically?
3. Can you explain how to find the sum of an arithmetic sequence in a general form?
4. What is the process of finding inserted arithmetic means?
5. How can you determine whether a given sequence is arithmetic, geometric, or harmonic?

Tip: Always check if a sequence is arithmetic, geometric, or harmonic by analyzing the relationships between consecutive terms.