For every integer n23 I+ 2+ 2?+ +4^= 4(4n-16) %3D 3. Proof (by induction)
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
![**Mathematical Induction Proof for a Summation Formula**
This document presents a proof by induction for a mathematical statement that holds for every integer \( n \geq 3 \).
**Statement:**
For every integer \( n \geq 3 \), the sum:
\[ 1 + 2 + 2^2 + \ldots + 4^n \]
is equal to:
\[ \frac{4(4^n - 16)}{3} \]
**Proof Outline**:
The proof is likely structured as a mathematical induction. Here's a typical approach:
1. **Base Case**: Verify the statement for the initial integer \( n = 3 \).
2. **Inductive Step**: Assume the statement holds for an arbitrary integer \( n = k \). Then prove it also holds for \( n = k + 1 \).
3. **Conclusion**: Conclude that the statement is true for all integers \( n \geq 3 \) based on the base case and inductive step.
This foundational method shows how each subsequent step can be deduced from the previous one, establishing the general validity of the formula.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2863cec8-c019-46a6-ad68-be87d1cedb5d%2Faa7d5314-a7ed-454d-92c7-a447352f20ca%2Fl7bw6zr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Induction Proof for a Summation Formula**
This document presents a proof by induction for a mathematical statement that holds for every integer \( n \geq 3 \).
**Statement:**
For every integer \( n \geq 3 \), the sum:
\[ 1 + 2 + 2^2 + \ldots + 4^n \]
is equal to:
\[ \frac{4(4^n - 16)}{3} \]
**Proof Outline**:
The proof is likely structured as a mathematical induction. Here's a typical approach:
1. **Base Case**: Verify the statement for the initial integer \( n = 3 \).
2. **Inductive Step**: Assume the statement holds for an arbitrary integer \( n = k \). Then prove it also holds for \( n = k + 1 \).
3. **Conclusion**: Conclude that the statement is true for all integers \( n \geq 3 \) based on the base case and inductive step.
This foundational method shows how each subsequent step can be deduced from the previous one, establishing the general validity of the formula.
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
