Geometric Sequence Worksheets - Free Printable

To solve the problems related to geometric sequences, we need to understand the definition and properties of a geometric sequence. A sequence is geometric if the ratio between consecutive terms is constant. This ratio is called the common ratio ($r$).

Problem 1: Which of the following is a geometric progression?

#### Options:
1. $25, 37.5, 56.25, \ldots$
2. $9.2, 27.8, 82.4, \ldots$
3. $5, 5\sqrt{2}, 7, \ldots$

Solution:
- Option 1: $25, 37.5, 56.25, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{37.5}{25} = 1.5, \quad \frac{56.25}{37.5} = 1.5
$$
The ratio is constant ($r = 1.5$), so this is a geometric progression.

- Option 2: $9.2, 27.8, 82.4, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{27.8}{9.2} \approx 3.0217, \quad \frac{82.4}{27.8} \approx 2.964
$$
The ratio is not constant, so this is not a geometric progression.

- Option 3: $5, 5\sqrt{2}, 7, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{5\sqrt{2}}{5} = \sqrt{2}, \quad \frac{7}{5\sqrt{2}} = \frac{7}{5\sqrt{2}} \neq \sqrt{2}
$$
The ratio is not constant, so this is not a geometric progression.

Answer:
$$
\boxed{1}
$$

Problem 2: Which of the following is a geometric sequence?

#### Options:
1. $11, 100, 900, \ldots$
2. $1, 1, 1, 1, \ldots$
3. $1, 2, 3, 4, \ldots$

Solution:
- Option 1: $11, 100, 900, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{100}{11} \approx 9.09, \quad \frac{900}{100} = 9
$$
The ratio is not constant, so this is not a geometric sequence.

- Option 2: $1, 1, 1, 1, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{1}{1} = 1, \quad \frac{1}{1} = 1
$$
The ratio is constant ($r = 1$), so this is a geometric sequence.

- Option 3: $1, 2, 3, 4, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{2}{1} = 2, \quad \frac{3}{2} = 1.5
$$
The ratio is not constant, so this is not a geometric sequence.

Answer:
$$
\boxed{2}
$$

Problem 3: Effect of the following on a geometric progression?

#### Options:
1. $(a, ar, ar^2, \ldots)$
2. $(ar, ar^2, ar^3, \ldots)$
3. $(ar^2, ar^3, ar^4, \ldots)$

Solution:
- Option 1: $(a, ar, ar^2, \ldots)$
- This is the standard form of a geometric progression with the first term $a$ and common ratio $r$.

- Option 2: $(ar, ar^2, ar^3, \ldots)$
- This is also a geometric progression, but it starts with the second term of the original sequence. The first term is $ar$, and the common ratio is still $r$.

- Option 3: $(ar^2, ar^3, ar^4, \ldots)$
- This is a geometric progression starting with the third term of the original sequence. The first term is $ar^2$, and the common ratio is still $r$.

All three options represent geometric progressions, but they start at different points in the sequence.

Answer:
$$
\boxed{1}
$$

Problem 4: Which of the following is a geometric sequence?

#### Options:
1. $7, 21, 63, \ldots$
2. $8, 16, 24, \ldots$
3. $6, 12, 24, \ldots$

Solution:
- Option 1: $7, 21, 63, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{21}{7} = 3, \quad \frac{63}{21} = 3
$$
The ratio is constant ($r = 3$), so this is a geometric sequence.

- Option 2: $8, 16, 24, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{16}{8} = 2, \quad \frac{24}{16} = 1.5
$$
The ratio is not constant, so this is not a geometric sequence.

- Option 3: $6, 12, 24, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{12}{6} = 2, \quad \frac{24}{12} = 2
$$
The ratio is constant ($r = 2$), so this is a geometric sequence.

Both Option 1 and Option 3 are geometric sequences, but since we need to choose one, we can select either. However, typically, the first correct option is chosen.

Answer:
$$
\boxed{1}
$$

Problem 5: Which of the following is not a geometric progression?

#### Options:
1. $1, 1, 1, 1, \ldots$
2. $1, 2, 4, 8, \ldots$
3. $1, 3, 5, 7, \ldots$

Solution:
- Option 1: $1, 1, 1, 1, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{1}{1} = 1, \quad \frac{1}{1} = 1
$$
The ratio is constant ($r = 1$), so this is a geometric progression.

- Option 2: $1, 2, 4, 8, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{2}{1} = 2, \quad \frac{4}{2} = 2
$$
The ratio is constant ($r = 2$), so this is a geometric progression.

- Option 3: $1, 3, 5, 7, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{3}{1} = 3, \quad \frac{5}{3} \approx 1.67
$$
The ratio is not constant, so this is not a geometric progression.

Answer:
$$
\boxed{3}
$$

Problem 6: Which of the following is a geometric progression?

#### Options:
1. $13, 26, 52, \ldots$
2. $13, 26, 39, \ldots$
3. $13, 26, 49, \ldots$

Solution:
- Option 1: $13, 26, 52, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{26}{13} = 2, \quad \frac{52}{26} = 2
$$
The ratio is constant ($r = 2$), so this is a geometric progression.

- Option 2: $13, 26, 39, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{26}{13} = 2, \quad \frac{39}{26} = 1.5
$$
The ratio is not constant, so this is not a geometric progression.

- Option 3: $13, 26, 49, \ldots$
- Check the ratio between consecutive terms:
$$
\frac{26}{13} = 2, \quad \frac{49}{26} \approx 1.88
$$
The ratio is not constant, so this is not a geometric progression.

Answer:
$$
\boxed{1}
$$

Final Answers:

1. $\boxed{1}$
2. $\boxed{2}$
3. $\boxed{1}$
4. $\boxed{1}$
5. $\boxed{3}$
6. $\boxed{1}$