Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Problem 3 Use Mathematical Induction to prove that 1² + 3² + 5²+...+ (2n + 1)² = (n + 1)(2n + 1) (2n + 3) 3 whenever n is a nonnegative integer. Solution 2

Problem 3 Use Mathematical Induction to prove that 1² + 3² + 5²+...+ (2n + 1)² = (n + 1)(2n + 1) (2n + 3) 3 whenever n is a nonnegative integer. Solution 2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
### Mathematical Induction Proof of Sum of Squares Formula #### Problem 3 **Use Mathematical Induction to prove that** \[ 1^2 + 3^2 + 5^2 + \cdots + (2n + 1)^2 = \frac{(n + 1)(2n + 1)(2n + 3)}{3} \] **whenever \( n \) is a nonnegative integer.** #### Solution: \[ \begin{aligned} & \text{Base case: Let } n = 0 \\ & (2 \cdot 0 + 1)^2 = \frac{(0 + 1)(2 \cdot 0 + 1)(2 \cdot 0 + 3)}{3} \\ & 1^2 = \frac{(1)(1)(3)}{3} \\ & 1 = 1 \text{, which is true.} \end{aligned} \] Next step is to assume the statement is true for some \( k \) and then prove it for \( k+1 \). This will be the inductive step. \[ \text{Assume } \sum_{i=0}^k (2i + 1)^2 = \frac{(k + 1)(2k + 1)(2k + 3)}{3} \] Then we need to show: \[ \sum_{i=0}^{k+1} (2i + 1)^2 = \frac{(k + 2)(2(k + 1) + 1)(2(k + 1) + 3)}{3} \] Add the next term \( (2(k+1) + 1)^2 \) to both sides of the assumed statement and simplify. Completing the details of the induction step ultimately demonstrates that the formula works for \( k+1 \), and by induction, it holds for all nonnegative integers \( n \). Thus, we have used mathematical induction to prove the given formula.
Transcribed Image Text:### Mathematical Induction Proof of Sum of Squares Formula #### Problem 3 **Use Mathematical Induction to prove that** \[ 1^2 + 3^2 + 5^2 + \cdots + (2n + 1)^2 = \frac{(n + 1)(2n + 1)(2n + 3)}{3} \] **whenever \( n \) is a nonnegative integer.** #### Solution: \[ \begin{aligned} & \text{Base case: Let } n = 0 \\ & (2 \cdot 0 + 1)^2 = \frac{(0 + 1)(2 \cdot 0 + 1)(2 \cdot 0 + 3)}{3} \\ & 1^2 = \frac{(1)(1)(3)}{3} \\ & 1 = 1 \text{, which is true.} \end{aligned} \] Next step is to assume the statement is true for some \( k \) and then prove it for \( k+1 \). This will be the inductive step. \[ \text{Assume } \sum_{i=0}^k (2i + 1)^2 = \frac{(k + 1)(2k + 1)(2k + 3)}{3} \] Then we need to show: \[ \sum_{i=0}^{k+1} (2i + 1)^2 = \frac{(k + 2)(2(k + 1) + 1)(2(k + 1) + 3)}{3} \] Add the next term \( (2(k+1) + 1)^2 \) to both sides of the assumed statement and simplify. Completing the details of the induction step ultimately demonstrates that the formula works for \( k+1 \), and by induction, it holds for all nonnegative integers \( n \). Thus, we have used mathematical induction to prove the given formula.
Expert Solution
steps

Step by step

Solved in 4 steps with 10 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS