Worksheet on identifying arithmetic and geometric sequences and writing their explicit formulas.

Let's solve the problem step by step.

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Part 1: Determine if each sequence is Arithmetic (A) or Geometric (G), and find the common difference or ratio.

We are given a table with sequences, and we need to fill in whether it's Arithmetic (A) or Geometric (G) and identify the common difference or common ratio.

Let’s go through each row:

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#### 1. Sequence: 1, 3, 5, 7, ...

- Differences:
3 - 1 = 2
5 - 3 = 2
7 - 5 = 2 → Common difference = +2
- This is an arithmetic sequence.
- ✔ Answer: A, +2

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#### 2. Sequence: 24, 12, 6, 3, ...

- Ratios:
12 / 24 = 0.5
6 / 12 = 0.5
3 / 6 = 0.5 → Common ratio = ×½
- This is a geometric sequence.
- ✔ Answer: G, ×½

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#### 3. Sequence: 7, 11, 15, 19, ...

- Differences:
11 - 7 = 4
15 - 11 = 4
19 - 15 = 4 → Common difference = +4
- This is arithmetic.
- ✔ Answer: A, +4

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#### 4. Sequence: 1, 4, 16, 64, ...

- Ratios:
4 / 1 = 4
16 / 4 = 4
64 / 16 = 4 → Common ratio = ×4
- This is geometric.
- ✔ Answer: G, ×4

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#### 5. Sequence: 27, 24, 21, 18, ...

- Differences:
24 - 27 = -3
21 - 24 = -3
18 - 21 = -3 → Common difference = -3
- This is arithmetic.
- ✔ Answer: A, -3

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#### 6. Sequence: ½, 1, 2, 4, ...

- Ratios:
1 / (½) = 2
2 / 1 = 2
4 / 2 = 2 → Common ratio = ×2
- This is geometric.
- ✔ Answer: G, ×2

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#### 7. Sequence: -6, 18, -54, 162, ...

- Ratios:
18 / (-6) = -3
-54 / 18 = -3
162 / (-54) = -3 → Common ratio = ×(-3)
- This is geometric.
- ✔ Answer: G, ×(-3)

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✔ So the completed table is:

| Sequence | Arithmetic (A) or Geometric (G) | Common Difference or Ratio |
|---------|-------------------------------|-----------------------------|
| 1, 3, 5, 7, ... | A | +2 |
| 24, 12, 6, 3, ... | G | ×½ |
| 7, 11, 15, 19, ... | A | +4 |
| 1, 4, 16, 64, ... | G | ×4 |
| 27, 24, 21, 18, ... | A | -3 |
| ½, 1, 2, 4, ... | G | ×2 |
| -6, 18, -54, 162, ... | G | ×(-3) |

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Part 2: Write the explicit formula for the following sequences

We use:
- Arithmetic: $ a_n = a_1 + (n-1)d $
- Geometric: $ a_n = a_1 \cdot r^{n-1} $

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#### 1) -4, -8, -12, -16, ...

- First term: $ a_1 = -4 $
- Common difference: $ d = -8 - (-4) = -4 $
- Formula:
$ a_n = -4 + (n-1)(-4) $
$ a_n = -4 -4(n-1) $
$ a_n = -4n $

✔ Answer: $ a_n = -4n $

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#### 2) 8, 6, 4, 2, ...

- First term: $ a_1 = 8 $
- Common difference: $ d = 6 - 8 = -2 $
- Formula:
$ a_n = 8 + (n-1)(-2) $
$ a_n = 8 - 2(n-1) $

✔ Answer: $ a_n = 8 - 2(n-1) $

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#### 3) 1, 3, 9, 27, ...

- First term: $ a_1 = 1 $
- Common ratio: $ r = 3/1 = 3 $
- Formula:
$ a_n = 1 \cdot 3^{n-1} $
$ a_n = 3^{n-1} $

✔ Answer: $ a_n = 3^{n-1} $

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#### 4) -20, -10, -5, -2.5, ...

- First term: $ a_1 = -20 $
- Common ratio: $ r = (-10)/(-20) = 0.5 $
- Formula:
$ a_n = -20 \cdot (0.5)^{n-1} $
$ a_n = -20 \left(\frac{1}{2}\right)^{n-1} $

✔ Answer: $ a_n = -20 \left(\frac{1}{2}\right)^{n-1} $

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#### 5) 10, 0, -10, -20, ...

- First term: $ a_1 = 10 $
- Common difference: $ d = 0 - 10 = -10 $
- Formula:
$ a_n = 10 + (n-1)(-10) $
$ a_n = 10 - 10(n-1) $

✔ Answer: $ a_n = 10 - 10(n-1) $

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#### 6) 9, 3, 1, \frac{1}{3}, ...

- First term: $ a_1 = 9 $
- Common ratio: $ r = 3/9 = 1/3 $
- Formula:
$ a_n = 9 \cdot \left(\frac{1}{3}\right)^{n-1} $

✔ Answer: $ a_n = 9 \left(\frac{1}{3}\right)^{n-1} $

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✔ Final Answers:

#### Table Completion:

| Sequence | Arithmetic (A) or Geometric (G) | Common Difference or Ratio |
|--------|-------------------------------|-----------------------------|
| 1, 3, 5, 7, ... | A | +2 |
| 24, 12, 6, 3, ... | G | ×½ |
| 7, 11, 15, 19, ... | A | +4 |
| 1, 4, 16, 64, ... | G | ×4 |
| 27, 24, 21, 18, ... | A | -3 |
| ½, 1, 2, 4, ... | G | ×2 |
| -6, 18, -54, 162, ... | G | ×(-3) |

#### Explicit Formulas:

1) $ a_n = -4n $
2) $ a_n = 8 - 2(n-1) $
3) $ a_n = 3^{n-1} $
4) $ a_n = -20 \left(\frac{1}{2}\right)^{n-1} $
5) $ a_n = 10 - 10(n-1) $
6) $ a_n = 9 \left(\frac{1}{3}\right)^{n-1} $

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Let me know if you'd like these formulas simplified further or explained in words!