Arithmetic and Geometric Sequence Worksheet for students to identify sequence types, calculate common differences or ratios, and find the next two terms.

Problem Analysis:

The task involves identifying whether each given sequence is arithmetic or geometric, determining the common difference (for arithmetic sequences) or the common ratio (for geometric sequences), and finding the next two terms of the sequence.

#### Definitions:
1. Arithmetic Sequence: A sequence where the difference between consecutive terms is constant.
- Common Difference: \( d = a_{n+1} - a_n \)
2. Geometric Sequence: A sequence where the ratio between consecutive terms is constant.
- Common Ratio: \( r = \frac{a_{n+1}}{a_n} \)

Solution:

#### Sequence 1: \( 14, 21, 28, 35, \ldots \)
- Check for Arithmetic Sequence:
\[
21 - 14 = 7, \quad 28 - 21 = 7, \quad 35 - 28 = 7
\]
The difference is constant (\( d = 7 \)). Therefore, it is an arithmetic sequence.
- Common Difference: \( d = 7 \)
- Next Two Terms:
\[
35 + 7 = 42, \quad 42 + 7 = 49
\]
So, the next two terms are \( 42 \) and \( 49 \).

#### Sequence 2: \( -1, 6, -36, 216, \ldots \)
- Check for Geometric Sequence:
\[
\frac{6}{-1} = -6, \quad \frac{-36}{6} = -6, \quad \frac{216}{-36} = -6
\]
The ratio is constant (\( r = -6 \)). Therefore, it is a geometric sequence.
- Common Ratio: \( r = -6 \)
- Next Two Terms:
\[
216 \cdot (-6) = -1296, \quad -1296 \cdot (-6) = 7776
\]
So, the next two terms are \( -1296 \) and \( 7776 \).

#### Sequence 3: \( -1, -5, -25, -125, \ldots \)
- Check for Geometric Sequence:
\[
\frac{-5}{-1} = 5, \quad \frac{-25}{-5} = 5, \quad \frac{-125}{-25} = 5
\]
The ratio is constant (\( r = 5 \)). Therefore, it is a geometric sequence.
- Common Ratio: \( r = 5 \)
- Next Two Terms:
\[
-125 \cdot 5 = -625, \quad -625 \cdot 5 = -3125
\]
So, the next two terms are \( -625 \) and \( -3125 \).

#### Sequence 4: \( 8, 2, -4, -10, \ldots \)
- Check for Arithmetic Sequence:
\[
2 - 8 = -6, \quad -4 - 2 = -6, \quad -10 - (-4) = -6
\]
The difference is constant (\( d = -6 \)). Therefore, it is an arithmetic sequence.
- Common Difference: \( d = -6 \)
- Next Two Terms:
\[
-10 + (-6) = -16, \quad -16 + (-6) = -22
\]
So, the next two terms are \( -16 \) and \( -22 \).

#### Sequence 5: \( 512, 128, 32, \ldots \)
- Check for Geometric Sequence:
\[
\frac{128}{512} = \frac{1}{4}, \quad \frac{32}{128} = \frac{1}{4}
\]
The ratio is constant (\( r = \frac{1}{4} \)). Therefore, it is a geometric sequence.
- Common Ratio: \( r = \frac{1}{4} \)
- Next Two Terms:
\[
32 \cdot \frac{1}{4} = 8, \quad 8 \cdot \frac{1}{4} = 2
\]
So, the next two terms are \( 8 \) and \( 2 \).

Final Answer:

\[
\boxed{
\begin{array}{|c|c|c|c|}
\hline
\text{Given Sequence} & \text{Arithmetic/Geometric} & \text{Common Difference/Common Ratio} & \text{Next Two Terms} \\
\hline
1) \ 14, 21, 28, 35, \ldots & \text{Arithmetic} & d = 7 & 42, 49 \\
\hline
2) \ -1, 6, -36, 216, \ldots & \text{Geometric} & r = -6 & -1296, 7776 \\
\hline
3) \ -1, -5, -25, -125, \ldots & \text{Geometric} & r = 5 & -625, -3125 \\
\hline
4) \ 8, 2, -4, -10, \ldots & \text{Arithmetic} & d = -6 & -16, -22 \\
\hline
5) \ 512, 128, 32, \ldots & \text{Geometric} & r = \frac{1}{4} & 8, 2 \\
\hline
\end{array}
}
\]