https://file.aiquickdraw.com/mathbot/file/17293361623805106lenz.jpg

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**.

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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.

Setelah mempelajari video, tentukan apakah soal di bawah ini termasuk barisan dan deret aritmatika atau barisan dan deret geometri. Kemudian tentukan 3 nilai (suku) selanjutnya.

Analyze the given sequences to identify if they are arithmetic or geometric, and find the next three terms. Includes examples for both sequence types, using formulas for common difference and common ratio. Suitable for Grades 8-10.

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