Arithmetic Sequences - Free Printable

Problem Overview:

The task involves identifying the next three terms and the common difference for each given arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant, known as the common difference.

Solution Approach:

1. Identify the Common Difference:
- The common difference (\(d\)) is calculated by subtracting any term from the next term in the sequence.
- \( d = a_{n+1} - a_n \)

2. Find the Next Three Terms:
- Once the common difference is identified, add it to the last term of the sequence to find the next term.
- Repeat this process to find the subsequent terms.

Step-by-Step Solutions:

#### Sequence 1: \(12, 24, 36, 48, 60, \_\_, \_\_, \_\_ \)
- Common Difference:
\[
d = 24 - 12 = 12
\]
- Next Three Terms:
\[
60 + 12 = 72, \quad 72 + 12 = 84, \quad 84 + 12 = 96
\]
- Answer: \(72, 84, 96\), Common Difference: \(12\)

#### Sequence 2: \(3, 0, -3, -6, -9, \_\_, \_\_, \_\_ \)
- Common Difference:
\[
d = 0 - 3 = -3
\]
- Next Three Terms:
\[
-9 + (-3) = -12, \quad -12 + (-3) = -15, \quad -15 + (-3) = -18
\]
- Answer: \(-12, -15, -18\), Common Difference: \(-3\)

#### Sequence 3: \(-15, -10, -5, 0, 5, \_\_, \_\_, \_\_ \)
- Common Difference:
\[
d = -10 - (-15) = 5
\]
- Next Three Terms:
\[
5 + 5 = 10, \quad 10 + 5 = 15, \quad 15 + 5 = 20
\]
- Answer: \(10, 15, 20\), Common Difference: \(5\)

#### Sequence 4: \(4.2, 4.5, 4.8, 5.1, 5.4, \_\_, \_\_, \_\_ \)
- Common Difference:
\[
d = 4.5 - 4.2 = 0.3
\]
- Next Three Terms:
\[
5.4 + 0.3 = 5.7, \quad 5.7 + 0.3 = 6.0, \quad 6.0 + 0.3 = 6.3
\]
- Answer: \(5.7, 6.0, 6.3\), Common Difference: \(0.3\)

#### Sequence 5: \(101, 159, 217, 275, 333, \_\_, \_\_, \_\_ \)
- Common Difference:
\[
d = 159 - 101 = 58
\]
- Next Three Terms:
\[
333 + 58 = 391, \quad 391 + 58 = 449, \quad 449 + 58 = 507
\]
- Answer: \(391, 449, 507\), Common Difference: \(58\)

#### Sequence 6: \(-50.2, -29.6, -9, 11.6, 32.2, \_\_, \_\_, \_\_ \)
- Common Difference:
\[
d = -29.6 - (-50.2) = 20.6
\]
- Next Three Terms:
\[
32.2 + 20.6 = 52.8, \quad 52.8 + 20.6 = 73.4, \quad 73.4 + 20.6 = 94
\]
- Answer: \(52.8, 73.4, 94\), Common Difference: \(20.6\)

#### Sequence 7: \(-784, -725, -666, -607, -548, \_\_, \_\_, \_\_ \)
- Common Difference:
\[
d = -725 - (-784) = 59
\]
- Next Three Terms:
\[
-548 + 59 = -489, \quad -489 + 59 = -430, \quad -430 + 59 = -371
\]
- Answer: \(-489, -430, -371\), Common Difference: \(59\)

#### Sequence 8: \(125, 83, 41, -1, -43, \_\_, \_\_, \_\_ \)
- Common Difference:
\[
d = 83 - 125 = -42
\]
- Next Three Terms:
\[
-43 + (-42) = -85, \quad -85 + (-42) = -127, \quad -127 + (-42) = -169
\]
- Answer: \(-85, -127, -169\), Common Difference: \(-42\)

Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{Sequence 1: } & 72, 84, 96, \text{ Common Difference: } 12 \\
\text{Sequence 2: } & -12, -15, -18, \text{ Common Difference: } -3 \\
\text{Sequence 3: } & 10, 15, 20, \text{ Common Difference: } 5 \\
\text{Sequence 4: } & 5.7, 6.0, 6.3, \text{ Common Difference: } 0.3 \\
\text{Sequence 5: } & 391, 449, 507, \text{ Common Difference: } 58 \\
\text{Sequence 6: } & 52.8, 73.4, 94, \text{ Common Difference: } 20.6 \\
\text{Sequence 7: } & -489, -430, -371, \text{ Common Difference: } 59 \\
\text{Sequence 8: } & -85, -127, -169, \text{ Common Difference: } -42 \\
\end{array}
}
\]