This worksheet helps students practice finding the nth term formula for various arithmetic sequences, including those with negative numbers and decimals.

Let's solve this step-by-step. The task is to find the nth term of each arithmetic sequence.

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🔍 What is an Arithmetic Sequence?

An arithmetic sequence has a constant difference between consecutive terms.
The nth term formula is:

$$
a_n = a + (n - 1)d
$$

Where:
- $ a $ = first term
- $ d $ = common difference
- $ n $ = position of the term

We can simplify this into the form:
$$
\text{nth term} = dn + c
$$
where $ c $ is adjusted so that when $ n = 1 $, we get the first term.

Alternatively, we often write it as:
$$
\boxed{a_n = dn + (a - d)}
$$
But more commonly, we derive it by finding $ d $ and then solving for the constant.

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## ✔ Section A: Finding nth Terms

We’ll go through each sequence.

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1. 4, 7, 10, 13

- $ a = 4 $
- $ d = 7 - 4 = 3 $
- So, $ a_n = 3n + b $
- When $ n = 1 $, $ a_1 = 3(1) + b = 4 \Rightarrow b = 1 $
- ✔ Answer: $ 3n + 1 $

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2. 6, 11, 16, 21

- $ a = 6 $
- $ d = 5 $
- $ a_n = 5n + b $
- $ n=1 $: $ 5(1) + b = 6 \Rightarrow b = 1 $
- ✔ $ 5n + 1 $

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3. 9, 11, 13, 15

- $ d = 2 $
- $ a = 9 $
- $ a_n = 2n + b $
- $ 2(1) + b = 9 \Rightarrow b = 7 $
- ✔ $ 2n + 7 $

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4. 15, 19, 23, 27

- $ d = 4 $
- $ a = 15 $
- $ a_n = 4n + b $
- $ 4(1) + b = 15 \Rightarrow b = 11 $
- ✔ $ 4n + 11 $

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5. 32, 42, 52, 62

- $ d = 10 $
- $ a = 32 $
- $ a_n = 10n + b $
- $ 10(1) + b = 32 \Rightarrow b = 22 $
- ✔ $ 10n + 22 $

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6. 76, 82, 88, 94

- $ d = 6 $
- $ a = 76 $
- $ a_n = 6n + b $
- $ 6(1) + b = 76 \Rightarrow b = 70 $
- ✔ $ 6n + 70 $

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7. 1, 4, 7, 10

- $ d = 3 $
- $ a = 1 $
- $ a_n = 3n + b $
- $ 3(1) + b = 1 \Rightarrow b = -2 $
- ✔ $ 3n - 2 $

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8. 3, 8, 13, 18

- $ d = 5 $
- $ a = 3 $
- $ a_n = 5n + b $
- $ 5(1) + b = 3 \Rightarrow b = -2 $
- ✔ $ 5n - 2 $

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9. -2, 0, 2, 4

- $ d = 2 $
- $ a = -2 $
- $ a_n = 2n + b $
- $ 2(1) + b = -2 \Rightarrow b = -4 $
- ✔ $ 2n - 4 $

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10. -5, -2, 1, 4

- $ d = 3 $
- $ a = -5 $
- $ a_n = 3n + b $
- $ 3(1) + b = -5 \Rightarrow b = -8 $
- ✔ $ 3n - 8 $

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11. -6, -1, 4, 9

- $ d = 5 $
- $ a = -6 $
- $ a_n = 5n + b $
- $ 5(1) + b = -6 \Rightarrow b = -11 $
- ✔ $ 5n - 11 $

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12. -10, -7, -4, -1

- $ d = 3 $
- $ a = -10 $
- $ a_n = 3n + b $
- $ 3(1) + b = -10 \Rightarrow b = -13 $
- ✔ $ 3n - 13 $

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## ✔ Section B: More Sequences

Now sequences with negative or decimal differences.

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1. 4, 3, 2, 1

- $ d = -1 $
- $ a = 4 $
- $ a_n = -n + b $
- $ -1(1) + b = 4 \Rightarrow b = 5 $
- ✔ $ -n + 5 $ or $ 5 - n $

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2. 8, 6, 4, 2

- $ d = -2 $
- $ a = 8 $
- $ a_n = -2n + b $
- $ -2(1) + b = 8 \Rightarrow b = 10 $
- ✔ $ -2n + 10 $

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3. 17, 12, 7, 2

- $ d = -5 $
- $ a = 17 $
- $ a_n = -5n + b $
- $ -5(1) + b = 17 \Rightarrow b = 22 $
- ✔ $ -5n + 22 $

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4. 0, -1, -2, -3

- $ d = -1 $
- $ a = 0 $
- $ a_n = -n + b $
- $ -1 + b = 0 \Rightarrow b = 1 $
- ✔ $ -n + 1 $ or $ 1 - n $

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5. 1, -1, -3, -5

- $ d = -2 $
- $ a = 1 $
- $ a_n = -2n + b $
- $ -2(1) + b = 1 \Rightarrow b = 3 $
- ✔ $ -2n + 3 $

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6. 3, -2, -7, -12

- $ d = -5 $
- $ a = 3 $
- $ a_n = -5n + b $
- $ -5(1) + b = 3 \Rightarrow b = 8 $
- ✔ $ -5n + 8 $

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7. -4, -7, -10, -13

- $ d = -3 $
- $ a = -4 $
- $ a_n = -3n + b $
- $ -3(1) + b = -4 \Rightarrow b = -1 $
- ✔ $ -3n - 1 $

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8. -11, -13, -15, -17

- $ d = -2 $
- $ a = -11 $
- $ a_n = -2n + b $
- $ -2(1) + b = -11 \Rightarrow b = -9 $
- ✔ $ -2n - 9 $

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9. -16, -20, -24, -28

- $ d = -4 $
- $ a = -16 $
- $ a_n = -4n + b $
- $ -4(1) + b = -16 \Rightarrow b = -12 $
- ✔ $ -4n - 12 $

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10. 1.5, 3.5, 5.5, 7.5

- $ d = 2 $
- $ a = 1.5 $
- $ a_n = 2n + b $
- $ 2(1) + b = 1.5 \Rightarrow b = -0.5 $
- ✔ $ 2n - 0.5 $

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11. 8.8, 8.6, 8.4, 8.2

- $ d = -0.2 $
- $ a = 8.8 $
- $ a_n = -0.2n + b $
- $ -0.2(1) + b = 8.8 \Rightarrow b = 9 $
- ✔ $ -0.2n + 9 $

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12. 3/4, 1/2, 1/4, 0

Convert to decimals: $ 0.75, 0.5, 0.25, 0 $

- $ d = -0.25 $
- $ a = 0.75 $
- $ a_n = -0.25n + b $
- $ -0.25(1) + b = 0.75 \Rightarrow b = 1 $
- ✔ $ -0.25n + 1 $

Or in fractions:
- $ d = -\frac{1}{4} $
- $ a = \frac{3}{4} $
- $ a_n = -\frac{1}{4}n + b $
- $ -\frac{1}{4}(1) + b = \frac{3}{4} \Rightarrow b = 1 $
- ✔ $ -\frac{1}{4}n + 1 $

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## 🌟 Extension: Create Sequences from Given Forms

We need to create sequences where the nth term is in one of these forms:

i. $ an + b $
ii. $ an - b $
iii. $ -an + b $
iv. $ -an - b $

With $ a $ and $ b $ being whole numbers.

Let’s pick simple values.

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i. $ an + b $

Let $ a = 2 $, $ b = 3 $ → $ 2n + 3 $

Sequence:
- $ n=1 $: $ 5 $
- $ n=2 $: $ 7 $
- $ n=3 $: $ 9 $
- $ n=4 $: $ 11 $

✔ Sequence: 5, 7, 9, 11

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ii. $ an - b $

Let $ a = 3 $, $ b = 1 $ → $ 3n - 1 $

- $ n=1 $: $ 2 $
- $ n=2 $: $ 5 $
- $ n=3 $: $ 8 $
- $ n=4 $: $ 11 $

✔ Sequence: 2, 5, 8, 11

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iii. $ -an + b $

Let $ a = 2 $, $ b = 5 $ → $ -2n + 5 $

- $ n=1 $: $ 3 $
- $ n=2 $: $ 1 $
- $ n=3 $: $ -1 $
- $ n=4 $: $ -3 $

✔ Sequence: 3, 1, -1, -3

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iv. $ -an - b $

Let $ a = 1 $, $ b = 2 $ → $ -n - 2 $

- $ n=1 $: $ -3 $
- $ n=2 $: $ -4 $
- $ n=3 $: $ -5 $
- $ n=4 $: $ -6 $

✔ Sequence: -3, -4, -5, -6

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## ✔ Final Answers Summary

Section A

| Sequence | nth Term |
|--------|----------|
| 4, 7, 10, 13 | $ 3n + 1 $ |
| 6, 11, 16, 21 | $ 5n + 1 $ |
| 9, 11, 13, 15 | $ 2n + 7 $ |
| 15, 19, 23, 27 | $ 4n + 11 $ |
| 32, 42, 52, 62 | $ 10n + 22 $ |
| 76, 82, 88, 94 | $ 6n + 70 $ |
| 1, 4, 7, 10 | $ 3n - 2 $ |
| 3, 8, 13, 18 | $ 5n - 2 $ |
| -2, 0, 2, 4 | $ 2n - 4 $ |
| -5, -2, 1, 4 | $ 3n - 8 $ |
| -6, -1, 4, 9 | $ 5n - 11 $ |
| -10, -7, -4, -1 | $ 3n - 13 $ |

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Section B

| Sequence | nth Term |
|--------|----------|
| 4, 3, 2, 1 | $ -n + 5 $ |
| 8, 6, 4, 2 | $ -2n + 10 $ |
| 17, 12, 7, 2 | $ -5n + 22 $ |
| 0, -1, -2, -3 | $ -n + 1 $ |
| 1, -1, -3, -5 | $ -2n + 3 $ |
| 3, -2, -7, -12 | $ -5n + 8 $ |
| -4, -7, -10, -13 | $ -3n - 1 $ |
| -11, -13, -15, -17 | $ -2n - 9 $ |
| -16, -20, -24, -28 | $ -4n - 12 $ |
| 1.5, 3.5, 5.5, 7.5 | $ 2n - 0.5 $ |
| 8.8, 8.6, 8.4, 8.2 | $ -0.2n + 9 $ |
| 3/4, 1/2, 1/4, 0 | $ -\frac{1}{4}n + 1 $ |

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Extension Examples

| Form | Example Sequence | nth Term |
|------|------------------|---------|
| i. $ an + b $ | 5, 7, 9, 11 | $ 2n + 3 $ |
| ii. $ an - b $ | 2, 5, 8, 11 | $ 3n - 1 $ |
| iii. $ -an + b $ | 3, 1, -1, -3 | $ -2n + 5 $ |
| iv. $ -an - b $ | -3, -4, -5, -6 | $ -n - 2 $ |

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