The question asks to determine whether the sequences provided are arithmetic or geometric and then to find the next three terms in each sequence. Let's go through each one.

1. 5, 9, 13, ...

   • This is an arithmetic sequence because the difference between consecutive terms is constant: $9 - 5 = 4,\quad 13 - 9 = 4.$ So, the common difference $d = 4$. The next three terms are: $13 + 4 = 17,\quad 17 + 4 = 21,\quad 21 + 4 = 25.$ Next terms: 17, 21, 25

2. -3, -5, -7, ...

   • This is also an arithmetic sequence because the difference between consecutive terms is constant: $-5 - (-3) = -2,\quad -7 - (-5) = -2.$ So, the common difference $d = -2$. The next three terms are: $-7 + (-2) = -9,\quad -9 + (-2) = -11,\quad -11 + (-2) = -13.$ Next terms: -9, -11, -13

3. 1, 4, 16, ...

   • This is a geometric sequence because each term is multiplied by a constant ratio: $\frac{4}{1} = 4,\quad \frac{16}{4} = 4.$ So, the common ratio $r = 4$. The next three terms are: $16 \times 4 = 64,\quad 64 \times 4 = 256,\quad 256 \times 4 = 1024.$ Next terms: 64, 256, 1024

4. -4, -1, 2, ...

   • This is an arithmetic sequence because the difference between consecutive terms is constant: $-1 - (-4) = 3,\quad 2 - (-1) = 3.$ So, the common difference $d = 3$. The next three terms are: $2 + 3 = 5,\quad 5 + 3 = 8,\quad 8 + 3 = 11.$ Next terms: 5, 8, 11

5. 20, 10, 5, ...

   • This is a geometric sequence because each term is multiplied by a constant ratio: $\frac{10}{20} = \frac{1}{2},\quad \frac{5}{10} = \frac{1}{2}.$ So, the common ratio $r = \frac{1}{2}$. The next three terms are: $5 \times \frac{1}{2} = 2.5,\quad 2.5 \times \frac{1}{2} = 1.25,\quad 1.25 \times \frac{1}{2} = 0.625.$ Next terms: 2.5, 1.25, 0.625

6. 2 + 4 + 6 + ...

   • This is an arithmetic series because the difference between consecutive terms is constant. The common difference is $2$, so the next terms are: $6 + 2 = 8,\quad 8 + 2 = 10,\quad 10 + 2 = 12.$ Next terms: 8, 10, 12

7. 6 + 3 + $\frac{3}{2}$ + ...

   • This is a geometric series because each term is divided by 2. The common ratio $r = \frac{1}{2}$. The next terms are: $\frac{3}{2} \times \frac{1}{2} = \frac{3}{4},\quad \frac{3}{4} \times \frac{1}{2} = \frac{3}{8},\quad \frac{3}{8} \times \frac{1}{2} = \frac{3}{16}.$ Next terms: $\frac{3}{4}, \frac{3}{8}, \frac{3}{16}$

8. -2 + 4 - 8 + ...

   • This is a geometric series with a common ratio of $-2$, as: $\frac{4}{-2} = -2,\quad \frac{-8}{4} = -2.$ The next three terms are: $-8 \times (-2) = 16,\quad 16 \times (-2) = -32,\quad -32 \times (-2) = 64.$ Next terms: 16, -32, 64

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the difference between an arithmetic and geometric sequence?
2. How do you calculate the $n$-th term of an arithmetic sequence?
3. Can a geometric sequence have a common ratio that is negative?
4. How do you find the sum of an arithmetic series?
5. What are some real-world applications of geometric sequences?

Tip: Always check the difference (for arithmetic) or ratio (for geometric) between terms to determine the sequence type.