Geometric Sequence worksheet - Free Printable

Problem Analysis and Solution

The worksheet focuses on Geometric Sequences. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

#### Part A: Identify the common ratio in each geometric sequence

We need to determine the common ratio \( r \) for each sequence by dividing any term by its preceding term.

1. Sequence: 3, 12, 48, 192
- Common ratio \( r = \frac{12}{3} = 4 \)
- Answer: \( r = 4 \)

2. Sequence: 64, 32, 16, 8
- Common ratio \( r = \frac{32}{64} = \frac{1}{2} \)
- Answer: \( r = \frac{1}{2} \)

3. Sequence: -1, 3, -9, 27
- Common ratio \( r = \frac{3}{-1} = -3 \)
- Answer: \( r = -3 \)

4. Sequence: \(\frac{1}{4}, \frac{1}{2}, 1, 2\)
- Common ratio \( r = \frac{\frac{1}{2}}{\frac{1}{4}} = 2 \)
- Answer: \( r = 2 \)

#### Part B: Find the nth term of the geometric sequence

The general formula for the \( n \)-th term of a geometric sequence is:
\[
g_n = g_1 \cdot r^{n-1}
\]
where:
- \( g_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

We will use this formula to solve each part.

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Problem 1: Sequence \( 2, -6, 18, -54, \ldots \)

- Given: \( g_1 = 2 \), \( r = -3 \)
- Find: \( g_7 \)

Using the formula \( g_n = g_1 \cdot r^{n-1} \):
\[
g_7 = g_1 \cdot r^{7-1} = 2 \cdot (-3)^6
\]

Calculate \( (-3)^6 \):
\[
(-3)^6 = 729
\]

Thus:
\[
g_7 = 2 \cdot 729 = 1458
\]

Answer: \( g_7 = 1458 \)

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Problem 2: Sequence \( 512, 128, 32, \ldots \)

- Given: \( g_1 = 512 \)
- Find: \( r \) and \( g_5 \)

#### Step 1: Find the common ratio \( r \)
\[
r = \frac{128}{512} = \frac{1}{4}
\]

#### Step 2: Find \( g_5 \)
Using the formula \( g_n = g_1 \cdot r^{n-1} \):
\[
g_5 = g_1 \cdot r^{5-1} = 512 \cdot \left(\frac{1}{4}\right)^4
\]

Calculate \( \left(\frac{1}{4}\right)^4 \):
\[
\left(\frac{1}{4}\right)^4 = \frac{1}{256}
\]

Thus:
\[
g_5 = 512 \cdot \frac{1}{256} = 2
\]

Answers: \( r = \frac{1}{4} \), \( g_5 = 2 \)

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Problem 3: Given \( g_2 = 5 \) and \( g_4 = 125 \)

- Find: \( g_1 \), \( r \), and \( g_5 \)

#### Step 1: Use the given terms to find \( r \)
The general formula for the \( n \)-th term is \( g_n = g_1 \cdot r^{n-1} \).

For \( g_2 \):
\[
g_2 = g_1 \cdot r^{2-1} = g_1 \cdot r = 5 \quad \text{(Equation 1)}
\]

For \( g_4 \):
\[
g_4 = g_1 \cdot r^{4-1} = g_1 \cdot r^3 = 125 \quad \text{(Equation 2)}
\]

#### Step 2: Solve for \( r \)
Divide Equation 2 by Equation 1:
\[
\frac{g_4}{g_2} = \frac{g_1 \cdot r^3}{g_1 \cdot r} = \frac{125}{5}
\]
\[
r^2 = 25
\]
\[
r = 5 \quad \text{(since \( r \) is positive in this context)}
\]

#### Step 3: Solve for \( g_1 \)
Substitute \( r = 5 \) into Equation 1:
\[
g_1 \cdot 5 = 5
\]
\[
g_1 = 1
\]

#### Step 4: Find \( g_5 \)
Using the formula \( g_n = g_1 \cdot r^{n-1} \):
\[
g_5 = g_1 \cdot r^{5-1} = 1 \cdot 5^4
\]

Calculate \( 5^4 \):
\[
5^4 = 625
\]

Thus:
\[
g_5 = 625
\]

Answers: \( g_1 = 1 \), \( r = 5 \), \( g_5 = 625 \)

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Final Answers

#### Part A:
1. \( r = 4 \)
2. \( r = \frac{1}{2} \)
3. \( r = -3 \)
4. \( r = 2 \)

#### Part B:
1. \( g_7 = 1458 \)
2. \( r = \frac{1}{4} \), \( g_5 = 2 \)
3. \( g_1 = 1 \), \( r = 5 \), \( g_5 = 625 \)

\boxed{1458, \frac{1}{4}, 2, 1, 5, 625}