**Fill in the missing terms for the geometric sequence.** | \( n \) | \( a_n \) | |---------|-----------| | 0 | | | 1 | 16 | | 2 | 64 | | 3 | | **Explanation:** You are given a table representing terms of a geometric sequence. The sequence starts from term \( n = 1 \) with a value of \( 16 \), and at \( n = 2 \), the value is \( 64 \). Your task is to find the missing terms for \( n = 0 \) and \( n = 3 \). To solve this, identify the common ratio, which you can calculate by dividing the term at \( n = 2 \) by the term at \( n = 1 \): \[ \text{Common ratio} = \frac{64}{16} = 4 \] Use this ratio to find the missing terms: - For \( n = 0 \): Divide \( a_1 \) by the common ratio. \[ a_0 = \frac{16}{4} = 4 \] - For \( n = 3 \): Multiply \( a_2 \) by the common ratio. \[ a_3 = 64 \times 4 = 256 \] Thus, the completed table should be: | \( n \) | \( a_n \) | |---------|-----------| | 0 | 4 | | 1 | 16 | | 2 | 64 | | 3 | 256 |
**Fill in the missing terms for the geometric sequence.** | \( n \) | \( a_n \) | |---------|-----------| | 0 | | | 1 | 16 | | 2 | 64 | | 3 | | **Explanation:** You are given a table representing terms of a geometric sequence. The sequence starts from term \( n = 1 \) with a value of \( 16 \), and at \( n = 2 \), the value is \( 64 \). Your task is to find the missing terms for \( n = 0 \) and \( n = 3 \). To solve this, identify the common ratio, which you can calculate by dividing the term at \( n = 2 \) by the term at \( n = 1 \): \[ \text{Common ratio} = \frac{64}{16} = 4 \] Use this ratio to find the missing terms: - For \( n = 0 \): Divide \( a_1 \) by the common ratio. \[ a_0 = \frac{16}{4} = 4 \] - For \( n = 3 \): Multiply \( a_2 \) by the common ratio. \[ a_3 = 64 \times 4 = 256 \] Thus, the completed table should be: | \( n \) | \( a_n \) | |---------|-----------| | 0 | 4 | | 1 | 16 | | 2 | 64 | | 3 | 256 |
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Fill in the missing terms for the geometric sequence.**
| \( n \) | \( a_n \) |
|---------|-----------|
| 0 | |
| 1 | 16 |
| 2 | 64 |
| 3 | |
**Explanation:**
You are given a table representing terms of a geometric sequence. The sequence starts from term \( n = 1 \) with a value of \( 16 \), and at \( n = 2 \), the value is \( 64 \). Your task is to find the missing terms for \( n = 0 \) and \( n = 3 \).
To solve this, identify the common ratio, which you can calculate by dividing the term at \( n = 2 \) by the term at \( n = 1 \):
\[ \text{Common ratio} = \frac{64}{16} = 4 \]
Use this ratio to find the missing terms:
- For \( n = 0 \): Divide \( a_1 \) by the common ratio.
\[ a_0 = \frac{16}{4} = 4 \]
- For \( n = 3 \): Multiply \( a_2 \) by the common ratio.
\[ a_3 = 64 \times 4 = 256 \]
Thus, the completed table should be:
| \( n \) | \( a_n \) |
|---------|-----------|
| 0 | 4 |
| 1 | 16 |
| 2 | 64 |
| 3 | 256 |](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfd0f760-2f1d-42e3-988d-506bce544ea6%2F7fe6bc35-4b1f-412d-93eb-b5b1a731c60b%2Fbtrsrcg_processed.png&w=3840&q=75)
Transcribed Image Text:**Fill in the missing terms for the geometric sequence.**
| \( n \) | \( a_n \) |
|---------|-----------|
| 0 | |
| 1 | 16 |
| 2 | 64 |
| 3 | |
**Explanation:**
You are given a table representing terms of a geometric sequence. The sequence starts from term \( n = 1 \) with a value of \( 16 \), and at \( n = 2 \), the value is \( 64 \). Your task is to find the missing terms for \( n = 0 \) and \( n = 3 \).
To solve this, identify the common ratio, which you can calculate by dividing the term at \( n = 2 \) by the term at \( n = 1 \):
\[ \text{Common ratio} = \frac{64}{16} = 4 \]
Use this ratio to find the missing terms:
- For \( n = 0 \): Divide \( a_1 \) by the common ratio.
\[ a_0 = \frac{16}{4} = 4 \]
- For \( n = 3 \): Multiply \( a_2 \) by the common ratio.
\[ a_3 = 64 \times 4 = 256 \]
Thus, the completed table should be:
| \( n \) | \( a_n \) |
|---------|-----------|
| 0 | 4 |
| 1 | 16 |
| 2 | 64 |
| 3 | 256 |
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