Algebra 2 Worksheets | Sequences and Series Worksheets - Free Printable

Let's solve each problem step by step and determine whether each sequence is arithmetic, geometric, or neither.

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🔹 Definitions:

- Arithmetic Sequence: Each term differs from the previous by a constant difference.
- Formula: $ a_n = a_1 + (n-1)d $
- Geometric Sequence: Each term is obtained by multiplying the previous term by a constant ratio.
- Formula: $ a_n = a_1 \cdot r^{n-1} $
- If neither condition holds, it's neither.

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Now let's analyze each:

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1) $ a_n = 5 - 3n $

This is a formula. Let’s compute first few terms:

- $ a_1 = 5 - 3(1) = 2 $
- $ a_2 = 5 - 3(2) = -1 $
- $ a_3 = 5 - 3(3) = -4 $

Sequence: 2, -1, -4, ...

Check differences:
- $ -1 - 2 = -3 $
- $ -4 - (-1) = -3 $

→ Constant difference of -3 → Arithmetic

✔ Answer: Arithmetic

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2) $ a_n = 5 \cdot 3^{n-4} $

Let’s find first few terms:

- $ a_1 = 5 \cdot 3^{1-4} = 5 \cdot 3^{-3} = 5 / 27 $
- $ a_2 = 5 \cdot 3^{-2} = 5 / 9 $
- $ a_3 = 5 \cdot 3^{-1} = 5 / 3 $
- $ a_4 = 5 \cdot 3^0 = 5 $

Sequence: $ \frac{5}{27}, \frac{5}{9}, \frac{5}{3}, 5, \dots $

Check ratios:
- $ \frac{5/9}{5/27} = \frac{5}{9} \cdot \frac{27}{5} = 3 $
- $ \frac{5/3}{5/9} = \frac{5}{3} \cdot \frac{9}{5} = 3 $
- $ \frac{5}{5/3} = 3 $

→ Common ratio = 3 → Geometric

✔ Answer: Geometric

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3) 3, 1, -1, -3, ...

Check differences:
- $ 1 - 3 = -2 $
- $ -1 - 1 = -2 $
- $ -3 - (-1) = -2 $

→ Constant difference of -2 → Arithmetic

✔ Answer: Arithmetic

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4) $ a_n = 2 \cdot 5^{n-4} $

Similar to #2.

- $ a_1 = 2 \cdot 5^{-3} = 2 / 125 $
- $ a_2 = 2 \cdot 5^{-2} = 2 / 25 $
- $ a_3 = 2 \cdot 5^{-1} = 2 / 5 $
- $ a_4 = 2 \cdot 5^0 = 2 $

Sequence: $ \frac{2}{125}, \frac{2}{25}, \frac{2}{5}, 2, \dots $

Ratios:
- $ \frac{2/25}{2/125} = \frac{2}{25} \cdot \frac{125}{2} = 5 $
- $ \frac{2/5}{2/25} = 5 $
- $ \frac{2}{2/5} = 5 $

→ Common ratio = 5 → Geometric

✔ Answer: Geometric

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5) 3, 10, 18, 34, ...

Check differences:
- $ 10 - 3 = 7 $
- $ 18 - 10 = 8 $
- $ 34 - 18 = 16 $

Not constant → Not arithmetic

Check ratios:
- $ 10 / 3 \approx 3.33 $
- $ 18 / 10 = 1.8 $
- $ 34 / 18 \approx 1.89 $

Not constant → Not geometric

✔ Answer: Neither

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6) $ a_n = 3 \cdot 5^{n-4} $

Again, similar pattern.

- $ a_1 = 3 \cdot 5^{-3} = 3 / 125 $
- $ a_2 = 3 / 25 $
- $ a_3 = 3 / 5 $
- $ a_4 = 3 $

Ratios:
- $ (3/25) / (3/125) = 5 $
- $ (3/5) / (3/25) = 5 $
- $ 3 / (3/5) = 5 $

→ Common ratio = 5 → Geometric

✔ Answer: Geometric

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7) $ a_n = 4 - 2n $

Compute terms:
- $ a_1 = 4 - 2(1) = 2 $
- $ a_2 = 4 - 4 = 0 $
- $ a_3 = 4 - 6 = -2 $
- $ a_4 = -4 $

Sequence: 2, 0, -2, -4, ...

Differences:
- $ 0 - 2 = -2 $
- $ -2 - 0 = -2 $
- $ -4 - (-2) = -2 $

→ Constant difference → Arithmetic

✔ Answer: Arithmetic

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8) 2, 6, 8, 18, ...

Check differences:
- $ 6 - 2 = 4 $
- $ 8 - 6 = 2 $
- $ 18 - 8 = 10 $

Not constant → Not arithmetic

Check ratios:
- $ 6 / 2 = 3 $
- $ 8 / 6 \approx 1.33 $
- $ 18 / 8 = 2.25 $

Not constant → Not geometric

✔ Answer: Neither

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9) 2, 6, 10, 14, ...

Differences:
- $ 6 - 2 = 4 $
- $ 10 - 6 = 4 $
- $ 14 - 10 = 4 $

→ Constant difference → Arithmetic

✔ Answer: Arithmetic

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10) $ a_n = 7 - 6^n $

Let’s compute terms:

- $ a_1 = 7 - 6^1 = 7 - 6 = 1 $
- $ a_2 = 7 - 6^2 = 7 - 36 = -29 $
- $ a_3 = 7 - 216 = -209 $
- $ a_4 = 7 - 1296 = -1289 $

Sequence: 1, -29, -209, -1289, ...

Differences:
- $ -29 - 1 = -30 $
- $ -209 - (-29) = -180 $
- $ -1289 - (-209) = -1080 $

Not constant → Not arithmetic

Ratios:
- $ -29 / 1 = -29 $
- $ -209 / -29 \approx 7.2 $
- $ -1289 / -209 \approx 6.17 $

Not constant → Not geometric

✔ Answer: Neither

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11) 1, 3, 4, 5, ...

Differences:
- $ 3 - 1 = 2 $
- $ 4 - 3 = 1 $
- $ 5 - 4 = 1 $

Not constant → Not arithmetic

Ratios:
- $ 3 / 1 = 3 $
- $ 4 / 3 \approx 1.33 $
- $ 5 / 4 = 1.25 $

Not constant → Not geometric

✔ Answer: Neither

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12) 4, 24, 144, 864, ...

Check ratios:
- $ 24 / 4 = 6 $
- $ 144 / 24 = 6 $
- $ 864 / 144 = 6 $

→ Common ratio = 6 → Geometric

✔ Answer: Geometric

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✔ Final Answers:

| Problem | Answer |
|--------|--------------|
| 1) | Arithmetic |
| 2) | Geometric |
| 3) | Arithmetic |
| 4) | Geometric |
| 5) | Neither |
| 6) | Geometric |
| 7) | Arithmetic |
| 8) | Neither |
| 9) | Arithmetic |
| 10) | Neither |
| 11) | Neither |
| 12) | Geometric |

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