Arithmetic Sequences Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem step by step from the worksheet titled "Arithmetic Sequences Practice 2".

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1. Is the sequence arithmetic or not?

$$
\{14, 24, 34, \ldots\}
$$

Solution:

An arithmetic sequence has a constant difference between consecutive terms.

Check the difference:
- $ 24 - 14 = 10 $
- $ 34 - 24 = 10 $

Since the common difference is constant ($d = 10$), this is an arithmetic sequence.

✔ Answer: Yes, it is arithmetic.

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2. Find the next 3 terms in the sequence.

$$
\left\{\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots\right\}
$$

Solution:

Look at the pattern:
- $ \frac{1}{2}, \frac{3}{2}, \frac{5}{2} $

The numerators are odd numbers increasing by 2: 1, 3, 5, ...

So the next three numerators are: 7, 9, 11

Thus, the next three terms are:
- $ \frac{7}{2}, \frac{9}{2}, \frac{11}{2} $

✔ Answer: $ \frac{7}{2}, \frac{9}{2}, \frac{11}{2} $

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5. Write a recursive rule for the $n$th term of the sequence:

$$
\{3.2, 4.7, 6.2, \ldots\}
$$

Solution:

First, find the common difference:
- $ 4.7 - 3.2 = 1.5 $
- $ 6.2 - 4.7 = 1.5 $

So $ d = 1.5 $

A recursive rule for an arithmetic sequence is:
$$
a_1 = \text{first term}, \quad a_n = a_{n-1} + d
$$

Here:
- $ a_1 = 3.2 $
- $ a_n = a_{n-1} + 1.5 $

✔ Answer:
$$
a_1 = 3.2, \quad a_n = a_{n-1} + 1.5
$$

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6. Write an explicit rule for the $n$th term of the sequence:

$$
\{-9, -17, -25, \ldots\}
$$
Then find $ a_{42} $

Solution:

Find the common difference:
- $ -17 - (-9) = -8 $
- $ -25 - (-17) = -8 $

So $ d = -8 $

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

Plug in:
- $ a_1 = -9 $
- $ d = -8 $

$$
a_n = -9 + (n - 1)(-8) = -9 -8(n - 1)
$$

Simplify:
$$
a_n = -9 -8n + 8 = -8n -1
$$

Now find $ a_{42} $:
$$
a_{42} = -8(42) - 1 = -336 - 1 = -337
$$

✔ Answer:
- Explicit rule: $ a_n = -8n - 1 $
- $ a_{42} = -337 $

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7. Write an explicit rule for the $n$th term of the sequence:

$$
\{1, 3, 5, 7, \ldots\}
$$
Then find $ a_{81} $

Solution:

This is the sequence of odd numbers.

Common difference: $ d = 2 $

First term: $ a_1 = 1 $

Explicit formula:
$$
a_n = 1 + (n - 1)(2) = 1 + 2n - 2 = 2n - 1
$$

Now find $ a_{81} $:
$$
a_{81} = 2(81) - 1 = 162 - 1 = 161
$$

✔ Answer:
- Explicit rule: $ a_n = 2n - 1 $
- $ a_{81} = 161 $

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8. Write a recursive rule for the $n$th term of the sequence:

$$
\{24, 30, 36, \ldots\}
$$
Then find $ a_7 $

Solution:

Common difference:
- $ 30 - 24 = 6 $
- $ 36 - 30 = 6 $

So $ d = 6 $

Recursive rule:
- $ a_1 = 24 $
- $ a_n = a_{n-1} + 6 $

Now compute up to $ a_7 $:

- $ a_1 = 24 $
- $ a_2 = 24 + 6 = 30 $
- $ a_3 = 30 + 6 = 36 $
- $ a_4 = 36 + 6 = 42 $
- $ a_5 = 42 + 6 = 48 $
- $ a_6 = 48 + 6 = 54 $
- $ a_7 = 54 + 6 = 60 $

✔ Answer:
- Recursive rule: $ a_1 = 24, \quad a_n = a_{n-1} + 6 $
- $ a_7 = 60 $

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9. Use the sequence to complete the table. Then graph it on the coordinate plane.

Sequence: $ \{-8, -5, -2, 1, 4, \ldots\} $

We are given:
| $ n $ | 1 | 2 | 3 | 4 | 5 | | | |
|--------|---|---|---|---|---|---|---|---|
| $ a(n) $ | -8 | -5 | -2 | 1 | 4 | | | |

Step 1: Find the common difference

- $ -5 - (-8) = 3 $
- $ -2 - (-5) = 3 $
- $ 1 - (-2) = 3 $
- $ 4 - 1 = 3 $

So $ d = 3 $

Next terms:
- $ a_6 = 4 + 3 = 7 $
- $ a_7 = 7 + 3 = 10 $
- $ a_8 = 10 + 3 = 13 $

Complete the table:

| $ n $ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|--------|---|---|---|---|---|---|---|---|
| $ a(n) $ | -8 | -5 | -2 | 1 | 4 | 7 | 10 | 13 |

Graphing:

Plot the points:
- (1, -8)
- (2, -5)
- (3, -2)
- (4, 1)
- (5, 4)
- (6, 7)
- (7, 10)
- (8, 13)

These points will form a straight line with slope $ m = 3 $, since it's an arithmetic sequence.

What do you notice about the graph?

✔ Answer: The graph is a straight line, which shows that the sequence is linear — consistent with arithmetic sequences.

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✔ Final Answers Summary:

1. Yes, it is arithmetic.
2. Next three terms: $ \frac{7}{2}, \frac{9}{2}, \frac{11}{2} $
5. Recursive rule: $ a_1 = 3.2, \quad a_n = a_{n-1} + 1.5 $
6. Explicit rule: $ a_n = -8n - 1 $, $ a_{42} = -337 $
7. Explicit rule: $ a_n = 2n - 1 $, $ a_{81} = 161 $
8. Recursive rule: $ a_1 = 24, \quad a_n = a_{n-1} + 6 $, $ a_7 = 60 $
9. Table completed: $ a(6)=7, a(7)=10, a(8)=13 $; Graph is a straight line.

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Let me know if you'd like a visual sketch of the graph or help formatting this for submission!