The task in the image asks to determine if the sequences given are arithmetic or geometric, and then to find the next three terms.

Let's analyze each sequence:

1. 5, 9, 13, ...

   • This is an arithmetic sequence because the difference between consecutive terms is constant: $9 - 5 = 13 - 9 = 4$.
   • The next three terms are obtained by adding 4 to the last term: $13 + 4 = 17,\quad 17 + 4 = 21,\quad 21 + 4 = 25.$
   • Next three terms: 17, 21, 25.

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

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

3. 1, 4, 16, ...

   • This is a geometric sequence because each term is multiplied by 4: $4 \div 1 = 4$ and $16 \div 4 = 4$.
   • The next three terms are: $16 \times 4 = 64,\quad 64 \times 4 = 256,\quad 256 \times 4 = 1024.$
   • Next three terms: 64, 256, 1024.

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

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

5. 20, 10, 5, ...

   • This is a geometric sequence because each term is divided by 2 (or multiplied by $\frac{1}{2}$): $10 \div 20 = \frac{1}{2}$ and $5 \div 10 = \frac{1}{2}$.
   • 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 three terms: 2.5, 1.25, 0.625.

6. 2 + 4 + 6 + ...

   • This is an arithmetic series with a common difference of 2.
   • The next three terms are: $6 + 2 = 8,\quad 8 + 2 = 10,\quad 10 + 2 = 12.$
   • Next three terms: 8, 10, 12.

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

   • This is a geometric series because each term is multiplied by $\frac{1}{2}$: $3 \div 6 = \frac{1}{2}$ and $\frac{3}{2} \div 3 = \frac{1}{2}$.
   • The next three 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 three terms: $\frac{3}{4}, \frac{3}{8}, \frac{3}{16}$.

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

   • This is a geometric series because each term is multiplied by -2: $4 \div (-2) = -2$ and $-8 \div 4 = -2$.
   • The next three terms are: $-8 \times (-2) = 16,\quad 16 \times (-2) = -32,\quad -32 \times (-2) = 64.$
   • Next three terms: 16, -32, 64.

Would you like more details or have any questions? Here are five related questions:

1. How can we derive the general formula for an arithmetic sequence?
2. What is the formula to find the sum of the first $n$ terms in a geometric series?
3. How can you identify the common ratio in a geometric sequence?
4. What happens when the common difference or ratio is negative in sequences?
5. How do we determine if a sequence is neither arithmetic nor geometric?

Tip: In an arithmetic sequence, the difference between consecutive terms is constant, while in a geometric sequence, each term is multiplied by a constant factor.