GEOMETRIC SEQUENCE AND SERIES WORKSHEET. The - Free Printable

Problem Analysis:

The task involves determining whether each given sequence is a geometric sequence, identifying the common ratio if it is geometric, and finding the next three terms. Additionally, there is a graphing task for one of the sequences and a question about the differences between geometric and arithmetic sequences.

Solution:

#### Step 1: Recall the Definition of a Geometric Sequence
A sequence is geometric if the ratio between consecutive terms is constant. This constant is called the common ratio ($r$). For a sequence $a_1, a_2, a_3, \ldots$, the common ratio is calculated as:
$$
r = \frac{a_{n+1}}{a_n}
$$
If the ratio is the same for all consecutive terms, the sequence is geometric.

#### Step 2: Solve Each Sequence

##### Sequence 1: $3, 6, 12, 24, \ldots$
- Check the ratio between consecutive terms:
$$
r = \frac{6}{3} = 2, \quad r = \frac{12}{6} = 2, \quad r = \frac{24}{12} = 2
$$
The ratio is constant ($r = 2$), so this is a geometric sequence.
- Next three terms:
$$
a_5 = 24 \cdot 2 = 48, \quad a_6 = 48 \cdot 2 = 96, \quad a_7 = 96 \cdot 2 = 192
$$
Answer: Geometric, $r = 2$, next three terms: $48, 96, 192$.

##### Sequence 2: $2, 6, 18, 54, \ldots$
- Check the ratio between consecutive terms:
$$
r = \frac{6}{2} = 3, \quad r = \frac{18}{6} = 3, \quad r = \frac{54}{18} = 3
$$
The ratio is constant ($r = 3$), so this is a geometric sequence.
- Next three terms:
$$
a_5 = 54 \cdot 3 = 162, \quad a_6 = 162 \cdot 3 = 486, \quad a_7 = 486 \cdot 3 = 1458
$$
Answer: Geometric, $r = 3$, next three terms: $162, 486, 1458$.

##### Sequence 3: $9, 3, 1, \frac{1}{3}, \ldots$
- Check the ratio between consecutive terms:
$$
r = \frac{3}{9} = \frac{1}{3}, \quad r = \frac{1}{3} = \frac{1}{3}, \quad r = \frac{\frac{1}{3}}{1} = \frac{1}{3}
$$
The ratio is constant ($r = \frac{1}{3}$), so this is a geometric sequence.
- Next three terms:
$$
a_5 = \frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}, \quad a_6 = \frac{1}{9} \cdot \frac{1}{3} = \frac{1}{27}, \quad a_7 = \frac{1}{27} \cdot \frac{1}{3} = \frac{1}{81}
$$
Answer: Geometric, $r = \frac{1}{3}$, next three terms: $\frac{1}{9}, \frac{1}{27}, \frac{1}{81}$.

##### Sequence 4: $2, 4, 6, 8, \ldots$
- Check the ratio between consecutive terms:
$$
r = \frac{4}{2} = 2, \quad r = \frac{6}{4} = 1.5, \quad r = \frac{8}{6} = \frac{4}{3}
$$
The ratio is not constant, so this is not a geometric sequence.
Answer: Not geometric.

##### Sequence 5: $1, -4, 16, -64, \ldots$
- Check the ratio between consecutive terms:
$$
r = \frac{-4}{1} = -4, \quad r = \frac{16}{-4} = -4, \quad r = \frac{-64}{16} = -4
$$
The ratio is constant ($r = -4$), so this is a geometric sequence.
- Next three terms:
$$
a_5 = -64 \cdot (-4) = 256, \quad a_6 = 256 \cdot (-4) = -1024, \quad a_7 = -1024 \cdot (-4) = 4096
$$
Answer: Geometric, $r = -4$, next three terms: $256, -1024, 4096$.

##### Sequence 6: $2, 3, 4.5, 6.75, \ldots$
- Check the ratio between consecutive terms:
$$
r = \frac{3}{2} = 1.5, \quad r = \frac{4.5}{3} = 1.5, \quad r = \frac{6.75}{4.5} = 1.5
$$
The ratio is constant ($r = 1.5$), so this is a geometric sequence.
- Next three terms:
$$
a_5 = 6.75 \cdot 1.5 = 10.125, \quad a_6 = 10.125 \cdot 1.5 = 15.1875, \quad a_7 = 15.1875 \cdot 1.5 = 22.78125
$$
Answer: Geometric, $r = 1.5$, next three terms: $10.125, 15.1875, 22.78125$.

##### Sequence 7: $10, 2, \frac{2}{5}, \frac{2}{25}, \ldots$
- Check the ratio between consecutive terms:
$$
r = \frac{2}{10} = \frac{1}{5}, \quad r = \frac{\frac{2}{5}}{2} = \frac{1}{5}, \quad r = \frac{\frac{2}{25}}{\frac{2}{5}} = \frac{1}{5}
$$
The ratio is constant ($r = \frac{1}{5}$), so this is a geometric sequence.
- Next three terms:
$$
a_5 = \frac{2}{25} \cdot \frac{1}{5} = \frac{2}{125}, \quad a_6 = \frac{2}{125} \cdot \frac{1}{5} = \frac{2}{625}, \quad a_7 = \frac{2}{625} \cdot \frac{1}{5} = \frac{2}{3125}
$$
Answer: Geometric, $r = \frac{1}{5}$, next three terms: $\frac{2}{125}, \frac{2}{625}, \frac{2}{3125}$.

##### Sequence 8: $6, 42, 294, \ldots$
- Check the ratio between consecutive terms:
$$
r = \frac{42}{6} = 7, \quad r = \frac{294}{42} = 7
$$
The ratio is constant ($r = 7$), so this is a geometric sequence.
- Next three terms:
$$
a_4 = 294 \cdot 7 = 2058, \quad a_5 = 2058 \cdot 7 = 14406, \quad a_6 = 14406 \cdot 7 = 100842
$$
Answer: Geometric, $r = 7$, next three terms: $2058, 14406, 100842$.

##### Sequence 9: $16, 20, 25, 31.25, \ldots$
- Check the ratio between consecutive terms:
$$
r = \frac{20}{16} = 1.25, \quad r = \frac{25}{20} = 1.25, \quad r = \frac{31.25}{25} = 1.25
$$
The ratio is constant ($r = 1.25$), so this is a geometric sequence.
- Next three terms:
$$
a_5 = 31.25 \cdot 1.25 = 39.0625, \quad a_6 = 39.0625 \cdot 1.25 = 48.828125, \quad a_7 = 48.828125 \cdot 1.25 = 61.03515625
$$
Answer: Geometric, $r = 1.25$, next three terms: $39.0625, 48.828125, 61.03515625$.

#### Step 3: Graph the Sequence in #6
The sequence in #6 is $2, 3, 4.5, 6.75, \ldots$ with $r = 1.5$. The terms can be plotted as points on a graph where the $x$-axis represents the term number and the $y$-axis represents the value of the term.

| Term Number ($n$) | Value ($a_n$) |
|-------------------|---------------|
| 1 | 2 |
| 2 | 3 |
| 3 | 4.5 |
| 4 | 6.75 |
| 5 | 10.125 |

Plot these points on the graph provided.

#### Step 4: Compare Geometric and Arithmetic Sequences
- Geometric Sequence: Each term is obtained by multiplying the previous term by a constant ratio ($r$). The graph of a geometric sequence is exponential (curved).
- Arithmetic Sequence: Each term is obtained by adding a constant difference ($d$) to the previous term. The graph of an arithmetic sequence is linear (straight line).

Final Answer:

\[
\boxed{
\begin{array}{ll}
1. & \text{Geometric, } r = 2, \text{ next three terms: } 48, 96, 192 \\
2. & \text{Geometric, } r = 3, \text{ next three terms: } 162, 486, 1458 \\
3. & \text{Geometric, } r = \frac{1}{3}, \text{ next three terms: } \frac{1}{9}, \frac{1}{27}, \frac{1}{81} \\
4. & \text{Not geometric} \\
5. & \text{Geometric, } r = -4, \text{ next three terms: } 256, -1024, 4096 \\
6. & \text{Geometric, } r = 1.5, \text{ next three terms: } 10.125, 15.1875, 22.78125 \\
7. & \text{Geometric, } r = \frac{1}{5}, \text{ next three terms: } \frac{2}{125}, \frac{2}{625}, \frac{2}{3125} \\
8. & \text{Geometric, } r = 7, \text{ next three terms: } 2058, 14406, 100842 \\
9. & \text{Geometric, } r = 1.25, \text{ next three terms: } 39.0625, 48.828125, 61.03515625 \\
\end{array}
}
\]