Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![Below is a description and transcription of the given image contents, formatted for use on an educational website.
---
**Mathematical Notation for Summation**
When working with sequences and series in mathematics, it is often practical to represent the sum of a sequence using sigma (Σ) notation. Here's how you might frame a particular problem using this notation:
### Example Problem:
**Given Sequence:**
\[ -3 - 6 - 12 + \ldots + 96 \]
**Task:**
Write the sum of this sequence using sigma notation.
**Solution:**
1. **Identify the Pattern:** Determine if the sequence is arithmetic or geometric. Here, we observe that the sequence is geometric with a common ratio (multiplication factor) of 2.
2. **General Term ( \(a_n\) ):** Find the nth term of the sequence. For this sequence, the nth term (\(a_n\)) can be written as:
\[ a_n = -3 \cdot 2^{(n-1)} \]
3. **Sigma Notation:** Using sigma notation, we can express the sum of the first n terms of this geometric sequence by:
\[ \sum_{n=1}^{N} -3 \cdot 2^{(n-1)} \]
Refer to the initial and final values in the sequence to set the bounds for the summation.
---
By understanding and using such mathematical notations, students can compactly and clearly represent complex sums, facilitating easier computation and comprehension.
**Graphical/Diagram Representation:**
- In the provided image, there is problem input termed:
\[\sum_{n=?} \]
This notation indicates the use of summation (sigma notation), but lacks specific bounds and terms, which are crucial to specify the exact sum.
This was an example of using sigma notation for representing a given sequence. Proper understanding of sigma notation can significantly enhance your ability to handle series and sequences in mathematics.
---
**Note:** Personal information has been redacted to emphasize educational content.
**Preview Button:**
In the given image, a 'Preview' button is visible. This likely allows the user to see the formatted output of their input before final submission.
---
Ensure to cross-verify each step in sum representations, especially while dealing with bounds and nth term formula derivation. Happy learning!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7921e840-b4be-4997-af5c-a45fa0acee7f%2F15861fd6-83bb-4c9b-946f-12240662f010%2F2dn0cy9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is a description and transcription of the given image contents, formatted for use on an educational website.
---
**Mathematical Notation for Summation**
When working with sequences and series in mathematics, it is often practical to represent the sum of a sequence using sigma (Σ) notation. Here's how you might frame a particular problem using this notation:
### Example Problem:
**Given Sequence:**
\[ -3 - 6 - 12 + \ldots + 96 \]
**Task:**
Write the sum of this sequence using sigma notation.
**Solution:**
1. **Identify the Pattern:** Determine if the sequence is arithmetic or geometric. Here, we observe that the sequence is geometric with a common ratio (multiplication factor) of 2.
2. **General Term ( \(a_n\) ):** Find the nth term of the sequence. For this sequence, the nth term (\(a_n\)) can be written as:
\[ a_n = -3 \cdot 2^{(n-1)} \]
3. **Sigma Notation:** Using sigma notation, we can express the sum of the first n terms of this geometric sequence by:
\[ \sum_{n=1}^{N} -3 \cdot 2^{(n-1)} \]
Refer to the initial and final values in the sequence to set the bounds for the summation.
---
By understanding and using such mathematical notations, students can compactly and clearly represent complex sums, facilitating easier computation and comprehension.
**Graphical/Diagram Representation:**
- In the provided image, there is problem input termed:
\[\sum_{n=?} \]
This notation indicates the use of summation (sigma notation), but lacks specific bounds and terms, which are crucial to specify the exact sum.
This was an example of using sigma notation for representing a given sequence. Proper understanding of sigma notation can significantly enhance your ability to handle series and sequences in mathematics.
---
**Note:** Personal information has been redacted to emphasize educational content.
**Preview Button:**
In the given image, a 'Preview' button is visible. This likely allows the user to see the formatted output of their input before final submission.
---
Ensure to cross-verify each step in sum representations, especially while dealing with bounds and nth term formula derivation. Happy learning!
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