12 (주) 3. 니.2 5k +2 K-1
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
I need help writing this in expanded form. (1st picture)
I need finding the actual value of the sum of this. (2nd picture).
![The image shows a mathematical expression involving a summation. The expression is as follows:
\[ 4 \cdot \sum_{k=1}^{12} \left( \frac{5k + 2}{4} \right)^3 \]
In this expression:
- The variable \( k \) is the index of summation.
- The summation ranges from \( k = 1 \) to \( k = 12 \).
- Inside the summation, the term is raised to the power of 3, with the expression \((\frac{5k + 2}{4})^3\).
- The entire summation is multiplied by 4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9acd99b6-ea84-4a4f-a75b-db40248206e3%2F17844e87-36d2-45cc-9bd2-d8f9f9aa3324%2F9uejiyl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical expression involving a summation. The expression is as follows:
\[ 4 \cdot \sum_{k=1}^{12} \left( \frac{5k + 2}{4} \right)^3 \]
In this expression:
- The variable \( k \) is the index of summation.
- The summation ranges from \( k = 1 \) to \( k = 12 \).
- Inside the summation, the term is raised to the power of 3, with the expression \((\frac{5k + 2}{4})^3\).
- The entire summation is multiplied by 4.
![The image contains handwritten mathematical notation with summation expressions.
1. **Expression on the Left:**
- Problem 3(b):
- The expression involves a summation:
\[
\sum_{k=1}^{n} \left\lfloor \frac{(12 + 3k)^3}{n} \right\rfloor
\]
- Here, \( n \) represents a variable, \( k \) is the index of summation starting at 1 and ending at \( n \). The expression inside the summation is the floor function applied to the cube of \((12 + 3k)\) divided by \( n \).
2. **Expression on the Right:**
- Problem 3(a):
- Another summation expression is shown:
\[
\sum_{i=2}^{4} 4 = 4 + 4 + 4
\]
- This illustrates a simple summation where the index \( i \) starts at 2 and ends at 4, repeatedly adding the number 4.
There are no graphs or diagrams in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9acd99b6-ea84-4a4f-a75b-db40248206e3%2F17844e87-36d2-45cc-9bd2-d8f9f9aa3324%2F5otswar_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains handwritten mathematical notation with summation expressions.
1. **Expression on the Left:**
- Problem 3(b):
- The expression involves a summation:
\[
\sum_{k=1}^{n} \left\lfloor \frac{(12 + 3k)^3}{n} \right\rfloor
\]
- Here, \( n \) represents a variable, \( k \) is the index of summation starting at 1 and ending at \( n \). The expression inside the summation is the floor function applied to the cube of \((12 + 3k)\) divided by \( n \).
2. **Expression on the Right:**
- Problem 3(a):
- Another summation expression is shown:
\[
\sum_{i=2}^{4} 4 = 4 + 4 + 4
\]
- This illustrates a simple summation where the index \( i \) starts at 2 and ends at 4, repeatedly adding the number 4.
There are no graphs or diagrams in the image.
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