Arithmetic Sequence Worksheets - Free Printable

You've uploaded a worksheet titled “General Term”, which has two parts:

- Part A: Given the general term formula (an), write out the arithmetic sequence.
- Part B: Given an arithmetic sequence, write the general term formula (an).

Let’s solve each problem step by step.

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## ✔ PART A: Write the arithmetic sequence using the given general term.

We’ll plug in n = 1, 2, 3, 4, ... into each formula to generate the first few terms.

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1) \( a_n = -23 + 10(n - 1) \)

This is already in the standard form: \( a_n = a_1 + (n-1)d \)

- \( a_1 = -23 + 10(0) = -23 \)
- \( a_2 = -23 + 10(1) = -13 \)
- \( a_3 = -23 + 10(2) = -3 \)
- \( a_4 = -23 + 10(3) = 7 \)

✔ Sequence: –23, –13, –3, 7, ...

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2) \( a_n = 7 - 6n \)

This is not in standard form, but we can still plug in values.

- \( a_1 = 7 - 6(1) = 1 \)
- \( a_2 = 7 - 6(2) = -5 \)
- \( a_3 = 7 - 6(3) = -11 \)
- \( a_4 = 7 - 6(4) = -17 \)

✔ Sequence: 1, –5, –11, –17, ...

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3) \( a_n = 15 - 5(n - 1) \)

- \( a_1 = 15 - 5(0) = 15 \)
- \( a_2 = 15 - 5(1) = 10 \)
- \( a_3 = 15 - 5(2) = 5 \)
- \( a_4 = 15 - 5(3) = 0 \)

✔ Sequence: 15, 10, 5, 0, ...

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4) \( a_n = \frac{23 + 8(n - 1)}{2} \)

Simplify or compute directly:

- \( a_1 = \frac{23 + 0}{2} = 11.5 \)
- \( a_2 = \frac{23 + 8}{2} = \frac{31}{2} = 15.5 \)
- \( a_3 = \frac{23 + 16}{2} = \frac{39}{2} = 19.5 \)
- \( a_4 = \frac{23 + 24}{2} = \frac{47}{2} = 23.5 \)

✔ Sequence: 11.5, 15.5, 19.5, 23.5, ...

*(Note: Decimals are fine — it’s still an arithmetic sequence with common difference 4)*

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5) \( a_n = -43 + 27n \)

- \( a_1 = -43 + 27(1) = -16 \)
- \( a_2 = -43 + 27(2) = 11 \)
- \( a_3 = -43 + 27(3) = 38 \)
- \( a_4 = -43 + 27(4) = 65 \)

✔ Sequence: –16, 11, 38, 65, ...

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6) \( a_n = 18 + 6(n + 1) \)

Wait — this is unusual. It uses \( (n+1) \), not \( (n-1) \). Let’s simplify:

\( a_n = 18 + 6(n + 1) = 18 + 6n + 6 = 24 + 6n \)

Now compute:

- \( a_1 = 24 + 6(1) = 30 \)
- \( a_2 = 24 + 6(2) = 36 \)
- \( a_3 = 24 + 6(3) = 42 \)
- \( a_4 = 24 + 6(4) = 48 \)

✔ Sequence: 30, 36, 42, 48, ...

*(Common difference = 6)*

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## ✔ PART B: Write the general term of each arithmetic sequence.

We use the formula:
> \( a_n = a_1 + (n - 1)d \)
where \( a_1 \) = first term, \( d \) = common difference.

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7) 78, 150, 222, 294, 366, ...

- \( a_1 = 78 \)
- \( d = 150 - 78 = 72 \)

✔ General term:
\( a_n = 78 + 72(n - 1) \)
*(Can also be simplified: \( a_n = 72n + 6 \))*

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8) 18, 25, 32, 39, 46, ...

- \( a_1 = 18 \)
- \( d = 25 - 18 = 7 \)

✔ General term:
\( a_n = 18 + 7(n - 1) \)
*(Simplified: \( a_n = 7n + 11 \))*

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9) –14, –30, –46, –62, –78, ...

- \( a_1 = -14 \)
- \( d = -30 - (-14) = -16 \)

✔ General term:
\( a_n = -14 + (-16)(n - 1) \)
or
\( a_n = -14 - 16(n - 1) \)
*(Simplified: \( a_n = -16n + 2 \))*

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10) 5, 8, 11, 14, 17, ...

- \( a_1 = 5 \)
- \( d = 8 - 5 = 3 \)

✔ General term:
\( a_n = 5 + 3(n - 1) \)
*(Simplified: \( a_n = 3n + 2 \))*

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## 📝 Final Answers:

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Part A: Sequences

1) –23, –13, –3, 7, ...
2) 1, –5, –11, –17, ...
3) 15, 10, 5, 0, ...
4) 11.5, 15.5, 19.5, 23.5, ...
5) –16, 11, 38, 65, ...
6) 30, 36, 42, 48, ...

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Part B: General Terms

7) \( a_n = 78 + 72(n - 1) \) or \( a_n = 72n + 6 \)
8) \( a_n = 18 + 7(n - 1) \) or \( a_n = 7n + 11 \)
9) \( a_n = -14 - 16(n - 1) \) or \( a_n = -16n + 2 \)
10) \( a_n = 5 + 3(n - 1) \) or \( a_n = 3n + 2 \)

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Let me know if you want these written in a specific format (like boxed answers or simplified forms only)!