Arithmetic And Geometric Sequences Coloring Activity - Free Printable

Problem Analysis

The task involves two main parts:
1. Identifying the type of sequence (arithmetic, geometric, or neither) for sequences 1 through 10.
2. Finding the next three terms in sequences 11 through 20.

Let's solve each part step by step.

---

Part 1: Identifying the Type of Sequence (Sequences 1–10)

#### Sequence 1: \( -3, -18, -108, -648, -3888, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
-18 - (-3) = -15, \quad -108 - (-18) = -90, \quad -648 - (-108) = -540
\]
The differences are not constant, so it is not arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{-18}{-3} = 6, \quad \frac{-108}{-18} = 6, \quad \frac{-648}{-108} = 6
\]
The ratios are constant (\( r = 6 \)), so it is geometric.

- Conclusion: Geometric

---

#### Sequence 2: \( 2, 4, 12, 48, 240, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
4 - 2 = 2, \quad 12 - 4 = 8, \quad 48 - 12 = 36
\]
The differences are not constant, so it is not arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{4}{2} = 2, \quad \frac{12}{4} = 3, \quad \frac{48}{12} = 4
\]
The ratios are not constant, so it is not geometric.

- Conclusion: Neither

---

#### Sequence 3: \( -35, 165, 365, 565, 765, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
165 - (-35) = 200, \quad 365 - 165 = 200, \quad 565 - 365 = 200
\]
The differences are constant (\( d = 200 \)), so it is arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{165}{-35} \neq \text{constant}
\]
The ratios are not constant, so it is not geometric.

- Conclusion: Arithmetic

---

#### Sequence 4: \( -2, 6, -18, 54, -162, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
6 - (-2) = 8, \quad -18 - 6 = -24, \quad 54 - (-18) = 72
\]
The differences are not constant, so it is not arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{6}{-2} = -3, \quad \frac{-18}{6} = -3, \quad \frac{54}{-18} = -3
\]
The ratios are constant (\( r = -3 \)), so it is geometric.

- Conclusion: Geometric

---

#### Sequence 5: \( -7, 93, 193, 293, 393, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
93 - (-7) = 100, \quad 193 - 93 = 100, \quad 293 - 193 = 100
\]
The differences are constant (\( d = 100 \)), so it is arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{93}{-7} \neq \text{constant}
\]
The ratios are not constant, so it is not geometric.

- Conclusion: Arithmetic

---

#### Sequence 6: \( 8, 14, 20, 26, 32, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
14 - 8 = 6, \quad 20 - 14 = 6, \quad 26 - 20 = 6
\]
The differences are constant (\( d = 6 \)), so it is arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{14}{8} \neq \text{constant}
\]
The ratios are not constant, so it is not geometric.

- Conclusion: Arithmetic

---

#### Sequence 7: \( -1, -2, -6, -24, -120, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
-2 - (-1) = -1, \quad -6 - (-2) = -4, \quad -24 - (-6) = -18
\]
The differences are not constant, so it is not arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{-2}{-1} = 2, \quad \frac{-6}{-2} = 3, \quad \frac{-24}{-6} = 4
\]
The ratios are not constant, so it is not geometric.

- Conclusion: Neither

---

#### Sequence 8: \( -18, -12, -6, 0, 6, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
-12 - (-18) = 6, \quad -6 - (-12) = 6, \quad 0 - (-6) = 6
\]
The differences are constant (\( d = 6 \)), so it is arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{-12}{-18} \neq \text{constant}
\]
The ratios are not constant, so it is not geometric.

- Conclusion: Arithmetic

---

#### Sequence 9: \( -2.5, -5, -10, -20, -40, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
-5 - (-2.5) = -2.5, \quad -10 - (-5) = -5, \quad -20 - (-10) = -10
\]
The differences are not constant, so it is not arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{-5}{-2.5} = 2, \quad \frac{-10}{-5} = 2, \quad \frac{-20}{-10} = 2
\]
The ratios are constant (\( r = 2 \)), so it is geometric.

- Conclusion: Geometric

---

#### Sequence 10: \( 1, 2, 6, 24, 120, \ldots \)

- Check for Arithmetic Sequence:
The difference between consecutive terms should be constant.
\[
2 - 1 = 1, \quad 6 - 2 = 4, \quad 24 - 6 = 18
\]
The differences are not constant, so it is not arithmetic.

- Check for Geometric Sequence:
The ratio between consecutive terms should be constant.
\[
\frac{2}{1} = 2, \quad \frac{6}{2} = 3, \quad \frac{24}{6} = 4
\]
The ratios are not constant, so it is not geometric.

- Conclusion: Neither

---

Summary for Part 1:

1. Geometric
2. Neither
3. Arithmetic
4. Geometric
5. Arithmetic
6. Arithmetic
7. Neither
8. Arithmetic
9. Geometric
10. Neither

---

Part 2: Finding the Next Three Terms (Sequences 11–20)

#### Sequence 11: \( 2, 4, 12, 48, 240, \ldots \)

- Pattern: Each term is multiplied by an increasing integer.
\[
2 \times 2 = 4, \quad 4 \times 3 = 12, \quad 12 \times 4 = 48, \quad 48 \times 5 = 240
\]
The next multipliers are 6, 7, and 8.
\[
240 \times 6 = 1440, \quad 1440 \times 7 = 10080, \quad 10080 \times 8 = 80640
\]

- Next Three Terms: \( 1440, 10080, 80640 \)

---

#### Sequence 12: \( 2, 5, 10, 17, 26, \ldots \)

- Pattern: The differences between consecutive terms increase by 2 each time.
\[
5 - 2 = 3, \quad 10 - 5 = 5, \quad 17 - 10 = 7, \quad 26 - 17 = 9
\]
The next differences are 11, 13, and 15.
\[
26 + 11 = 37, \quad 37 + 13 = 50, \quad 50 + 15 = 65
\]

- Next Three Terms: \( 37, 50, 65 \)

---

#### Sequence 13: \( 1, 9, 25, 49, 81, \ldots \)

- Pattern: The terms are squares of odd integers.
\[
1^2 = 1, \quad 3^2 = 9, \quad 5^2 = 25, \quad 7^2 = 49, \quad 9^2 = 81
\]
The next terms are \( 11^2, 13^2, 15^2 \).
\[
11^2 = 121, \quad 13^2 = 169, \quad 15^2 = 225
\]

- Next Three Terms: \( 121, 169, 225 \)

---

#### Sequence 14: \( 4, 16, 36, 64, 100, \ldots \)

- Pattern: The terms are squares of even integers.
\[
2^2 = 4, \quad 4^2 = 16, \quad 6^2 = 36, \quad 8^2 = 64, \quad 10^2 = 100
\]
The next terms are \( 12^2, 14^2, 16^2 \).
\[
12^2 = 144, \quad 14^2 = 196, \quad 16^2 = 256
\]

- Next Three Terms: \( 144, 196, 256 \)

---

#### Sequence 15: \( -6, -2, 0, 1, \frac{3}{2}, \ldots \)

- Pattern: The differences between consecutive terms form a sequence that increases by \( \frac{1}{2} \) each time.
\[
-2 - (-6) = 4, \quad 0 - (-2) = 2, \quad 1 - 0 = 1, \quad \frac{3}{2} - 1 = \frac{1}{2}
\]
The next differences are \( 0, -\frac{1}{2}, -1 \).
\[
\frac{3}{2} + 0 = \frac{3}{2}, \quad \frac{3}{2} + \left(-\frac{1}{2}\right) = 1, \quad 1 + (-1) = 0
\]

- Next Three Terms: \( \frac{3}{2}, 1, 0 \)

---

#### Sequence 16: \( -9, -7, -4, 0, 5, \ldots \)

- Pattern: The differences between consecutive terms increase by 2 each time.
\[
-7 - (-9) = 2, \quad -4 - (-7) = 3, \quad 0 - (-4) = 4, \quad 5 - 0 = 5
\]
The next differences are 6, 7, and 8.
\[
5 + 6 = 11, \quad 11 + 7 = 18, \quad 18 + 8 = 26
\]

- Next Three Terms: \( 11, 18, 26 \)

---

#### Sequence 17: \( -2, -4, -12, -48, -240, \ldots \)

- Pattern: Each term is multiplied by an increasing integer.
\[
-2 \times 2 = -4, \quad -4 \times 3 = -12, \quad -12 \times 4 = -48, \quad -48 \times 5 = -240
\]
The next multipliers are 6, 7, and 8.
\[
-240 \times 6 = -1440, \quad -1440 \times 7 = -10080, \quad -10080 \times 8 = -80640
\]

- Next Three Terms: \( -1440, -10080, -80640 \)

---

#### Sequence 18: \( 6, 8, 11, 15, 20, \ldots \)

- Pattern: The differences between consecutive terms increase by 1 each time.
\[
8 - 6 = 2, \quad 11 - 8 = 3, \quad 15 - 11 = 4, \quad 20 - 15 = 5
\]
The next differences are 6, 7, and 8.
\[
20 + 6 = 26, \quad 26 + 7 = 33, \quad 33 + 8 = 41
\]

- Next Three Terms: \( 26, 33, 41 \)

---

#### Sequence 19: \( -30, -14, -6, -2, 0, \ldots \)

- Pattern: The differences between consecutive terms decrease by 2 each time.
\[
-14 - (-30) = 16, \quad -6 - (-14) = 8, \quad -2 - (-6) = 4, \quad 0 - (-2) = 2
\]
The next differences are 0, -2, and -4.
\[
0 + 0 = 0, \quad 0 + (-2) = -2, \quad -2 + (-4) = -6
\]

- Next Three Terms: \( 0, -2, -6 \)

---

#### Sequence 20: \( -2, 1, 6, 13, 22, \ldots \)

- Pattern: The differences between consecutive terms increase by 2 each time.
\[
1 - (-2) = 3, \quad 6 - 1 = 5, \quad 13 - 6 = 7, \quad 22 - 13 = 9
\]
The next differences are 11, 13, and 15.
\[
22 + 11 = 33, \quad 33 + 13 = 46, \quad 46 + 15 = 61
\]

- Next Three Terms: \( 33, 46, 61 \)

---

Final Answers:

#### Part 1: Types of Sequences
1. Geometric
2. Neither
3. Arithmetic
4. Geometric
5. Arithmetic
6. Arithmetic
7. Neither
8. Arithmetic
9. Geometric
10. Neither

#### Part 2: Next Three Terms
11. \( 1440, 10080, 80640 \)
12. \( 37, 50, 65 \)
13. \( 121, 169, 225 \)
14. \( 144, 196, 256 \)
15. \( \frac{3}{2}, 1, 0 \)
16. \( 11, 18, 26 \)
17. \( -1440, -10080, -80640 \)
18. \( 26, 33, 41 \)
19. \( 0, -2, -6 \)
20. \( 33, 46, 61 \)

\boxed{
\begin{array}{ll}
\text{Part 1:} & \text{1. Geometric, 2. Neither, 3. Arithmetic, 4. Geometric, 5. Arithmetic, 6. Arithmetic, 7. Neither, 8. Arithmetic, 9. Geometric, 10. Neither} \\
\text{Part 2:} & \text{11. } 1440, 10080, 80640, \text{ 12. } 37, 50, 65, \text{ 13. } 121, 169, 225, \text{ 14. } 144, 196, 256, \\
& \text{15. } \frac{3}{2}, 1, 0, \text{ 16. } 11, 18, 26, \text{ 17. } -1440, -10080, -80640, \text{ 18. } 26, 33, 41, \\
& \text{19. } 0, -2, -6, \text{ 20. } 33, 46, 61
\end{array}
}