Comparing Arithmetic and Geometric Sequences Worksheet for 9th ... - Free Printable

Problem Analysis:

The task involves identifying whether each given sequence is arithmetic, geometric, or neither. Let's solve this step by step for each sequence.

---

Definitions:

1. Arithmetic Sequence: A sequence where the difference between consecutive terms is constant.
- If $ a_n $ is the $ n $-th term, then $ a_{n+1} - a_n = d $ (constant difference).
2. Geometric Sequence: A sequence where the ratio between consecutive terms is constant.
- If $ a_n $ is the $ n $-th term, then $ \frac{a_{n+1}}{a_n} = r $ (constant ratio).
3. Neither: If the sequence does not satisfy the conditions for either arithmetic or geometric.

---

Solutions:

#### Sequence 1: $ 1, 3, 6, 10, 15, \ldots $
- Check for arithmetic:
- Differences between consecutive terms:
$$
3 - 1 = 2, \quad 6 - 3 = 3, \quad 10 - 6 = 4, \quad 15 - 10 = 5
$$
- The differences are not constant ($ 2, 3, 4, 5 $), so it is not arithmetic.
- Check for geometric:
- Ratios between consecutive terms:
$$
\frac{3}{1} = 3, \quad \frac{6}{3} = 2, \quad \frac{10}{6} = \frac{5}{3}, \quad \frac{15}{10} = \frac{3}{2}
$$
- The ratios are not constant, so it is not geometric.
- Conclusion: This sequence is neither arithmetic nor geometric.

#### Sequence 2: $ 48, 41, 34, 27, 20, \ldots $
- Check for arithmetic:
- Differences between consecutive terms:
$$
41 - 48 = -7, \quad 34 - 41 = -7, \quad 27 - 34 = -7, \quad 20 - 27 = -7
$$
- The differences are constant ($ -7 $), so it is arithmetic.
- Check for geometric:
- Ratios between consecutive terms:
$$
\frac{41}{48}, \quad \frac{34}{41}, \quad \frac{27}{34}, \quad \frac{20}{27}
$$
- The ratios are not constant, so it is not geometric.
- Conclusion: This sequence is arithmetic.

#### Sequence 3: $ \frac{17}{9}, \frac{19}{9}, \frac{21}{9}, \frac{23}{9}, \frac{25}{9}, \ldots $
- Check for arithmetic:
- Differences between consecutive terms:
$$
\frac{19}{9} - \frac{17}{9} = \frac{2}{9}, \quad \frac{21}{9} - \frac{19}{9} = \frac{2}{9}, \quad \frac{23}{9} - \frac{21}{9} = \frac{2}{9}, \quad \frac{25}{9} - \frac{23}{9} = \frac{2}{9}
$$
- The differences are constant ($ \frac{2}{9} $), so it is arithmetic.
- Check for geometric:
- Ratios between consecutive terms:
$$
\frac{\frac{19}{9}}{\frac{17}{9}} = \frac{19}{17}, \quad \frac{\frac{21}{9}}{\frac{19}{9}} = \frac{21}{19}, \quad \frac{\frac{23}{9}}{\frac{21}{9}} = \frac{23}{21}, \quad \frac{\frac{25}{9}}{\frac{23}{9}} = \frac{25}{23}
$$
- The ratios are not constant, so it is not geometric.
- Conclusion: This sequence is arithmetic.

#### Sequence 4: $ -4, -12, -36, -108, \ldots $
- Check for arithmetic:
- Differences between consecutive terms:
$$
-12 - (-4) = -8, \quad -36 - (-12) = -24, \quad -108 - (-36) = -72
$$
- The differences are not constant ($ -8, -24, -72 $), so it is not arithmetic.
- Check for geometric:
- Ratios between consecutive terms:
$$
\frac{-12}{-4} = 3, \quad \frac{-36}{-12} = 3, \quad \frac{-108}{-36} = 3
$$
- The ratios are constant ($ 3 $), so it is geometric.
- Conclusion: This sequence is geometric.

#### Sequence 5: $ 6, 36, 216, 1296, \ldots $
- Check for arithmetic:
- Differences between consecutive terms:
$$
36 - 6 = 30, \quad 216 - 36 = 180, \quad 1296 - 216 = 1080
$$
- The differences are not constant ($ 30, 180, 1080 $), so it is not arithmetic.
- Check for geometric:
- Ratios between consecutive terms:
$$
\frac{36}{6} = 6, \quad \frac{216}{36} = 6, \quad \frac{1296}{216} = 6
$$
- The ratios are constant ($ 6 $), so it is geometric.
- Conclusion: This sequence is geometric.

#### Sequence 6: $ 1, 1, 2, 3, 5, \ldots $
- Check for arithmetic:
- Differences between consecutive terms:
$$
1 - 1 = 0, \quad 2 - 1 = 1, \quad 3 - 2 = 1, \quad 5 - 3 = 2
$$
- The differences are not constant ($ 0, 1, 1, 2 $), so it is not arithmetic.
- Check for geometric:
- Ratios between consecutive terms:
$$
\frac{1}{1} = 1, \quad \frac{2}{1} = 2, \quad \frac{3}{2} = \frac{3}{2}, \quad \frac{5}{3} = \frac{5}{3}
$$
- The ratios are not constant, so it is not geometric.
- Conclusion: This sequence is neither arithmetic nor geometric.

#### Sequence 7: $ -36, -28, -20, -12, \ldots $
- Check for arithmetic:
- Differences between consecutive terms:
$$
-28 - (-36) = 8, \quad -20 - (-28) = 8, \quad -12 - (-20) = 8
$$
- The differences are constant ($ 8 $), so it is arithmetic.
- Check for geometric:
- Ratios between consecutive terms:
$$
\frac{-28}{-36} = \frac{7}{9}, \quad \frac{-20}{-28} = \frac{5}{7}, \quad \frac{-12}{-20} = \frac{3}{5}
$$
- The ratios are not constant, so it is not geometric.
- Conclusion: This sequence is arithmetic.

#### Sequence 8: $ 9, 3, 1, \frac{1}{3}, \frac{1}{9}, \ldots $
- Check for arithmetic:
- Differences between consecutive terms:
$$
3 - 9 = -6, \quad 1 - 3 = -2, \quad \frac{1}{3} - 1 = -\frac{2}{3}, \quad \frac{1}{9} - \frac{1}{3} = -\frac{2}{9}
$$
- The differences are not constant, so it is not arithmetic.
- Check for geometric:
- Ratios between consecutive terms:
$$
\frac{3}{9} = \frac{1}{3}, \quad \frac{1}{3} = \frac{1}{3}, \quad \frac{\frac{1}{3}}{1} = \frac{1}{3}, \quad \frac{\frac{1}{9}}{\frac{1}{3}} = \frac{1}{3}
$$
- The ratios are constant ($ \frac{1}{3} $), so it is geometric.
- Conclusion: This sequence is geometric.

---

Final Answers:

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

$$
\boxed{\text{Neither, Arithmetic, Arithmetic, Geometric, Geometric, Neither, Arithmetic, Geometric}}
$$