Fill in the missing terms for the geometric sequence. n 1 2 3 an 16 64
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Title: Understanding Geometric Sequences**
**Objective:** Fill in the missing terms for the geometric sequence.
**Instructions:**
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.
**Given Table:**
\[
\begin{array}{|c|c|}
\hline
n & a_n \\
\hline
0 & \_\_ \\
1 & 16 \\
2 & 64 \\
3 & \_\_ \\
\hline
\end{array}
\]
**Steps to Solve:**
1. **Identify the Common Ratio:**
- To find the common ratio (\(r\)), divide the second term by the first term:
\[
r = \frac{64}{16} = 4
\]
2. **Find the Missing Terms:**
- **Term when \(n = 0\):**
- Since \(a_1 = a_0 \times r\), then:
\[
a_0 = \frac{16}{4} = 4
\]
- **Term when \(n = 3\):**
- Since \(a_3 = a_2 \times r\), then:
\[
a_3 = 64 \times 4 = 256
\]
3. **Complete Sequence:**
- \[
\begin{array}{|c|c|}
\hline
n & a_n \\
\hline
0 & 4 \\
1 & 16 \\
2 & 64 \\
3 & 256 \\
\hline
\end{array}
\]
**Conclusion:**
By understanding and applying the concept of a geometric sequence, you can determine any missing terms by utilizing the common ratio. This table provides a clear demonstration of the pattern and relationship between the terms in a geometric sequence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfd0f760-2f1d-42e3-988d-506bce544ea6%2Fe97a8ae8-dd14-4635-b6db-b100a1fc05d4%2F2a72b5p_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Geometric Sequences**
**Objective:** Fill in the missing terms for the geometric sequence.
**Instructions:**
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.
**Given Table:**
\[
\begin{array}{|c|c|}
\hline
n & a_n \\
\hline
0 & \_\_ \\
1 & 16 \\
2 & 64 \\
3 & \_\_ \\
\hline
\end{array}
\]
**Steps to Solve:**
1. **Identify the Common Ratio:**
- To find the common ratio (\(r\)), divide the second term by the first term:
\[
r = \frac{64}{16} = 4
\]
2. **Find the Missing Terms:**
- **Term when \(n = 0\):**
- Since \(a_1 = a_0 \times r\), then:
\[
a_0 = \frac{16}{4} = 4
\]
- **Term when \(n = 3\):**
- Since \(a_3 = a_2 \times r\), then:
\[
a_3 = 64 \times 4 = 256
\]
3. **Complete Sequence:**
- \[
\begin{array}{|c|c|}
\hline
n & a_n \\
\hline
0 & 4 \\
1 & 16 \\
2 & 64 \\
3 & 256 \\
\hline
\end{array}
\]
**Conclusion:**
By understanding and applying the concept of a geometric sequence, you can determine any missing terms by utilizing the common ratio. This table provides a clear demonstration of the pattern and relationship between the terms in a geometric sequence.
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