Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement:**
"Find the sum:"
\[ \sum_{k=2}^{5} (2^k - 1) \]
---
**Solution Box:**
[Blank space for answer]
---
**Need Help?**
- **Read It**: [Button for additional reading]
- **Watch It**: [Button for video explanation]
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**Explanation of the Mathematical Expression:**
The expression is a summation notation, which indicates the sum of a sequence. In this case, it is the sum of \(2^k - 1\) where \(k\) ranges from 2 to 5. Here's how to break it down:
1. **Sigma (\(\sum\)) Notation:** This symbol indicates that you will sum the expressions for each integer \(k\) starting from the lower limit (2) to the upper limit (5).
2. **Expression:**
- The term \(2^k\) represents 2 raised to the power of \(k\).
- Subtract 1 from this result for each \(k\).
3. **Calculate Terms:**
- For \(k=2\): \(2^2 - 1 = 4 - 1 = 3\)
- For \(k=3\): \(2^3 - 1 = 8 - 1 = 7\)
- For \(k=4\): \(2^4 - 1 = 16 - 1 = 15\)
- For \(k=5\): \(2^5 - 1 = 32 - 1 = 31\)
4. **Sum the Values:**
- \(3 + 7 + 15 + 31 = 56\)
The final sum is 56.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba53c783-81fd-4f77-a273-a15291fa98c2%2F1f3dafed-640c-45e8-9b05-5a82f52cb807%2Fm44osr9_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
"Find the sum:"
\[ \sum_{k=2}^{5} (2^k - 1) \]
---
**Solution Box:**
[Blank space for answer]
---
**Need Help?**
- **Read It**: [Button for additional reading]
- **Watch It**: [Button for video explanation]
---
**Explanation of the Mathematical Expression:**
The expression is a summation notation, which indicates the sum of a sequence. In this case, it is the sum of \(2^k - 1\) where \(k\) ranges from 2 to 5. Here's how to break it down:
1. **Sigma (\(\sum\)) Notation:** This symbol indicates that you will sum the expressions for each integer \(k\) starting from the lower limit (2) to the upper limit (5).
2. **Expression:**
- The term \(2^k\) represents 2 raised to the power of \(k\).
- Subtract 1 from this result for each \(k\).
3. **Calculate Terms:**
- For \(k=2\): \(2^2 - 1 = 4 - 1 = 3\)
- For \(k=3\): \(2^3 - 1 = 8 - 1 = 7\)
- For \(k=4\): \(2^4 - 1 = 16 - 1 = 15\)
- For \(k=5\): \(2^5 - 1 = 32 - 1 = 31\)
4. **Sum the Values:**
- \(3 + 7 + 15 + 31 = 56\)
The final sum is 56.
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