Practice worksheet for finding the nth term of sequences with examples and exercises.

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## 🔢 Question 1: Find the nth term for each sequence

All these are arithmetic sequences (constant difference between terms).

The general formula for the nth term of an arithmetic sequence is:

> aₙ = a + (n - 1)d

Where:
- `a` = first term
- `d` = common difference
- `n` = term number

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✔ (a) 5, 8, 11, 14, ...

- a = 5
- d = 8 - 5 = 3
- aₙ = 5 + (n - 1)×3 = 5 + 3n - 3 = 3n + 2

✔ Answer: 3n + 2

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✔ (b) 9, 14, 19, 24, ...

- a = 9
- d = 14 - 9 = 5
- aₙ = 9 + (n - 1)×5 = 9 + 5n - 5 = 5n + 4

✔ Answer: 5n + 4

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✔ (c) 1, 3, 5, 7, ...

- a = 1
- d = 2
- aₙ = 1 + (n - 1)×2 = 1 + 2n - 2 = 2n - 1

✔ Answer: 2n - 1

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✔ (d) 10, 14, 18, 22, ...

- a = 10
- d = 4
- aₙ = 10 + (n - 1)×4 = 10 + 4n - 4 = 4n + 6

✔ Answer: 4n + 6

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✔ (e) 2, 7, 12, 17, ...

- a = 2
- d = 5
- aₙ = 2 + (n - 1)×5 = 2 + 5n - 5 = 5n - 3

✔ Answer: 5n - 3

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✔ (f) 3, 9, 15, 21, ...

- a = 3
- d = 6
- aₙ = 3 + (n - 1)×6 = 3 + 6n - 6 = 6n - 3

✔ Answer: 6n - 3

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✔ (g) 11, 31, 51, 71, ...

- a = 11
- d = 20
- aₙ = 11 + (n - 1)×20 = 11 + 20n - 20 = 20n - 9

✔ Answer: 20n - 9

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✔ (h) 20, 23, 26, 29, ...

- a = 20
- d = 3
- aₙ = 20 + (n - 1)×3 = 20 + 3n - 3 = 3n + 17

✔ Answer: 3n + 17

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✔ (i) 1, 7, 13, 19, ...

- a = 1
- d = 6
- aₙ = 1 + (n - 1)×6 = 1 + 6n - 6 = 6n - 5

✔ Answer: 6n - 5

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✔ (j) 100, 125, 150, 175, ...

- a = 100
- d = 25
- aₙ = 100 + (n - 1)×25 = 100 + 25n - 25 = 25n + 75

✔ Answer: 25n + 75

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✔ (k) 13, 22, 31, 40, ...

- a = 13
- d = 9
- aₙ = 13 + (n - 1)×9 = 13 + 9n - 9 = 9n + 4

✔ Answer: 9n + 4

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✔ (l) 1.5, 2, 2.5, 3, ...

- a = 1.5
- d = 0.5
- aₙ = 1.5 + (n - 1)×0.5 = 1.5 + 0.5n - 0.5 = 0.5n + 1

✔ Answer: 0.5n + 1 (or ½n + 1)

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## 🔢 Question 2: Find the nth term for each sequence

These are also arithmetic sequences, but with negative or decreasing differences.

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✔ (a) 10, 7, 4, 1, ...

- a = 10
- d = 7 - 10 = -3
- aₙ = 10 + (n - 1)(-3) = 10 - 3n + 3 = -3n + 13

✔ Answer: -3n + 13

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✔ (b) 6, 4, 2, 0, ...

- a = 6
- d = -2
- aₙ = 6 + (n - 1)(-2) = 6 - 2n + 2 = -2n + 8

✔ Answer: -2n + 8

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✔ (c) 9, 4, -1, -6, ...

- a = 9
- d = -5
- aₙ = 9 + (n - 1)(-5) = 9 - 5n + 5 = -5n + 14

✔ Answer: -5n + 14

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✔ (d) 20, 10, 0, -10, ...

- a = 20
- d = -10
- aₙ = 20 + (n - 1)(-10) = 20 - 10n + 10 = -10n + 30

✔ Answer: -10n + 30

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✔ (e) 5, -1, -7, -13, ...

- a = 5
- d = -6
- aₙ = 5 + (n - 1)(-6) = 5 - 6n + 6 = -6n + 11

✔ Answer: -6n + 11

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✔ (f) 5, 4, 3, 2, ...

- a = 5
- d = -1
- aₙ = 5 + (n - 1)(-1) = 5 - n + 1 = -n + 6

✔ Answer: -n + 6

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✔ (g) -6, -13, -20, -27, ...

- a = -6
- d = -7
- aₙ = -6 + (n - 1)(-7) = -6 -7n + 7 = -7n + 1

✔ Answer: -7n + 1

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✔ (h) -10, -13, -16, -19, ...

- a = -10
- d = -3
- aₙ = -10 + (n - 1)(-3) = -10 -3n + 3 = -3n -7

✔ Answer: -3n - 7

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✔ (i) 2.5, 2, 1.5, 1, ...

- a = 2.5
- d = -0.5
- aₙ = 2.5 + (n - 1)(-0.5) = 2.5 - 0.5n + 0.5 = -0.5n + 3

✔ Answer: -0.5n + 3 (or -½n + 3)

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## 🎯 Final Summary:

Question 1 Answers:

| Part | nth Term |
|------|------------------|
| (a) | 3n + 2 |
| (b) | 5n + 4 |
| (c) | 2n - 1 |
| (d) | 4n + 6 |
| (e) | 5n - 3 |
| (f) | 6n - 3 |
| (g) | 20n - 9 |
| (h) | 3n + 17 |
| (i) | 6n - 5 |
| (j) | 25n + 75 |
| (k) | 9n + 4 |
| (l) | 0.5n + 1 |

Question 2 Answers:

| Part | nth Term |
|------|------------------|
| (a) | -3n + 13 |
| (b) | -2n + 8 |
| (c) | -5n + 14 |
| (d) | -10n + 30 |
| (e) | -6n + 11 |
| (f) | -n + 6 |
| (g) | -7n + 1 |
| (h) | -3n - 7 |
| (i) | -0.5n + 3 |

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✔ All sequences are linear (arithmetic) — so their nth terms are linear expressions in `n`.

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