9+ Arithmetic Sequence Examples - DOC, PDF, Excel - Free Printable

Let's solve each problem step by step.

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Part 1: Determine whether each sequence is arithmetic. If so, find the common difference.

An arithmetic sequence has a constant difference between consecutive terms. The common difference $ d $ is found by subtracting any term from the next term:
$$
d = a_{n+1} - a_n
$$

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#### 1) 15.2, 8.5, 1.8, -4.9 ...

Check differences:
- $ 8.5 - 15.2 = -6.7 $
- $ 1.8 - 8.5 = -6.7 $
- $ -4.9 - 1.8 = -6.7 $

✔ Common difference: $ d = -6.7 $ → Arithmetic

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#### 2) 8.4, 16.1, 23.8, 31.5 ...

Check differences:
- $ 16.1 - 8.4 = 7.7 $
- $ 23.8 - 16.1 = 7.7 $
- $ 31.5 - 23.8 = 7.7 $

✔ Common difference: $ d = 7.7 $ → Arithmetic

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#### 3) 5.1, -0.8, -6.7, -12.6 ...

Check differences:
- $ -0.8 - 5.1 = -5.9 $
- $ -6.7 - (-0.8) = -6.7 + 0.8 = -5.9 $
- $ -12.6 - (-6.7) = -12.6 + 6.7 = -5.9 $

✔ Common difference: $ d = -5.9 $ → Arithmetic

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#### 4) 14.3, 20.3, 26.3, 32.3 ...

Check differences:
- $ 20.3 - 14.3 = 6 $
- $ 26.3 - 20.3 = 6 $
- $ 32.3 - 26.3 = 6 $

✔ Common difference: $ d = 6 $ → Arithmetic

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✔ Summary of Part 1:

All four sequences are arithmetic.
- 1) $ d = -6.7 $
- 2) $ d = 7.7 $
- 3) $ d = -5.9 $
- 4) $ d = 6 $

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Part 2: Find the first four terms and stated term given the arithmetic sequence

Use the formula:
$$
a_n = a_1 + (n - 1)d
$$
But here, formulas are given directly in terms of $ n $. We can plug in values.

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#### 5) $ a_n = 25.5 - 6.8n $, find $ a_{10} $

We don't need to find $ a_1 $, just plug in $ n = 10 $:

$$
a_{10} = 25.5 - 6.8(10) = 25.5 - 68 = -42.5
$$

Now find first four terms:
- $ a_1 = 25.5 - 6.8(1) = 25.5 - 6.8 = 18.7 $
- $ a_2 = 25.5 - 6.8(2) = 25.5 - 13.6 = 11.9 $
- $ a_3 = 25.5 - 6.8(3) = 25.5 - 20.4 = 5.1 $
- $ a_4 = 25.5 - 6.8(4) = 25.5 - 27.2 = -1.7 $

✔ First four terms: 18.7, 11.9, 5.1, -1.7, $ a_{10} = -42.5 $

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#### 6) $ a_n = 25 - 10n $, find $ a_5 $

$$
a_5 = 25 - 10(5) = 25 - 50 = -25
$$

First four terms:
- $ a_1 = 25 - 10(1) = 15 $
- $ a_2 = 25 - 20 = 5 $
- $ a_3 = 25 - 30 = -5 $
- $ a_4 = 25 - 40 = -15 $

✔ First four terms: 15, 5, -5, -15, $ a_5 = -25 $

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#### 7) $ a_n = 11 + 9n $, find $ a_8 $

$$
a_8 = 11 + 9(8) = 11 + 72 = 83
$$

First four terms:
- $ a_1 = 11 + 9(1) = 20 $
- $ a_2 = 11 + 18 = 29 $
- $ a_3 = 11 + 27 = 38 $
- $ a_4 = 11 + 36 = 47 $

✔ First four terms: 20, 29, 38, 47, $ a_8 = 83 $

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#### 8) $ a_n = 65 - 35n $, find $ a_9 $

$$
a_9 = 65 - 35(9) = 65 - 315 = -250
$$

First four terms:
- $ a_1 = 65 - 35 = 30 $
- $ a_2 = 65 - 70 = -5 $
- $ a_3 = 65 - 105 = -40 $
- $ a_4 = 65 - 140 = -75 $

✔ First four terms: 30, -5, -40, -75, $ a_9 = -250 $

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✔ Summary of Part 2:

| Problem | First Four Terms | Stated Term |
|--------|------------------|-------------|
| 5) | 18.7, 11.9, 5.1, -1.7 | $ a_{10} = -42.5 $ |
| 6) | 15, 5, -5, -15 | $ a_5 = -25 $ |
| 7) | 20, 29, 38, 47 | $ a_8 = 83 $ |
| 8) | 30, -5, -40, -75 | $ a_9 = -250 $ |

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Part 3: Given $ a_1 $ and $ d $, find first four terms and formula

The general formula for an arithmetic sequence:
$$
a_n = a_1 + (n - 1)d
$$

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#### 9) $ a_1 = 25 $, $ d = 100 $

First four terms:
- $ a_1 = 25 $
- $ a_2 = 25 + 100 = 125 $
- $ a_3 = 125 + 100 = 225 $
- $ a_4 = 225 + 100 = 325 $

Formula:
$$
a_n = 25 + (n - 1)(100) = 25 + 100n - 100 = 100n - 75
$$

✔ First four: 25, 125, 225, 325, Formula: $ a_n = 100n - 75 $

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#### 10) $ a_1 = 5 $, $ d = 5 $

First four terms:
- $ a_1 = 5 $
- $ a_2 = 5 + 5 = 10 $
- $ a_3 = 10 + 5 = 15 $
- $ a_4 = 15 + 5 = 20 $

Formula:
$$
a_n = 5 + (n - 1)(5) = 5 + 5n - 5 = 5n
$$

✔ First four: 5, 10, 15, 20, Formula: $ a_n = 5n $

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#### 11) $ a_1 = 24 $, $ d = -15 $

First four terms:
- $ a_1 = 24 $
- $ a_2 = 24 - 15 = 9 $
- $ a_3 = 9 - 15 = -6 $
- $ a_4 = -6 - 15 = -21 $

Formula:
$$
a_n = 24 + (n - 1)(-15) = 24 - 15n + 15 = -15n + 39
$$

✔ First four: 24, 9, -6, -21, Formula: $ a_n = -15n + 39 $

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#### 12) $ a_1 = 9 $, $ d = -50 $

First four terms:
- $ a_1 = 9 $
- $ a_2 = 9 - 50 = -41 $
- $ a_3 = -41 - 50 = -91 $
- $ a_4 = -91 - 50 = -141 $

Formula:
$$
a_n = 9 + (n - 1)(-50) = 9 - 50n + 50 = -50n + 59
$$

✔ First four: 9, -41, -91, -141, Formula: $ a_n = -50n + 59 $

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✔ Final Answers:

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Part 1: Arithmetic? Find $ d $

1) Yes, $ d = -6.7 $
2) Yes, $ d = 7.7 $
3) Yes, $ d = -5.9 $
4) Yes, $ d = 6 $

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Part 2: First four terms & stated term

5) First four: 18.7, 11.9, 5.1, -1.7; $ a_{10} = -42.5 $
6) First four: 15, 5, -5, -15; $ a_5 = -25 $
7) First four: 20, 29, 38, 47; $ a_8 = 83 $
8) First four: 30, -5, -40, -75; $ a_9 = -250 $

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Part 3: Given $ a_1 $, $ d $ → first four terms & formula

9) First four: 25, 125, 225, 325; $ a_n = 100n - 75 $
10) First four: 5, 10, 15, 20; $ a_n = 5n $
11) First four: 24, 9, -6, -21; $ a_n = -15n + 39 $
12) First four: 9, -41, -91, -141; $ a_n = -50n + 59 $

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