Part A Here are three attempted statements of a theorem which was provided in the video (it is also Example 2.5.1 in DMOI). None of these statements are correct. For each one, explain which piece is incorrect, why it matters, and how to fix it. "Theorem 1": For all integers n E Z, "Theorem 2": For all natural numbers n > 1, "Theorem 3": For all natural numbers n > 1, i=0 i=0 n i=0 i n(n + 1) 2 i(i+1) 2 n(n-1) 2
Part A Here are three attempted statements of a theorem which was provided in the video (it is also Example 2.5.1 in DMOI). None of these statements are correct. For each one, explain which piece is incorrect, why it matters, and how to fix it. "Theorem 1": For all integers n E Z, "Theorem 2": For all natural numbers n > 1, "Theorem 3": For all natural numbers n > 1, i=0 i=0 n i=0 i n(n + 1) 2 i(i+1) 2 n(n-1) 2
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
Can you help me solve this induction problem?
![Part A
Here are three attempted statements of a theorem which was provided in the video (it is also Example 2.5.1 in DMOI). None of these statements are correct. For each one, explain which piece is incorrect, why it matters, and how to fix it.
“Theorem 1”: For all integers \( n \in \mathbb{Z} \),
\[
\sum_{i=0}^{n} i = \frac{n(n + 1)}{2}.
\]
“Theorem 2”: For all natural numbers \( n \geq 1 \),
\[
\sum_{i=0}^{n} i = \frac{i(i + 1)}{2}.
\]
“Theorem 3”: For all natural numbers \( n \geq 1 \),
\[
\sum_{i=0}^{n} i = \frac{n(n - 1)}{2}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F385eb8fb-6648-4df6-8047-4f830c46ad3b%2F885c38e7-2fad-418c-9f2d-c2e5866bf297%2Flgzkdoo_processed.png&w=3840&q=75)
Transcribed Image Text:Part A
Here are three attempted statements of a theorem which was provided in the video (it is also Example 2.5.1 in DMOI). None of these statements are correct. For each one, explain which piece is incorrect, why it matters, and how to fix it.
“Theorem 1”: For all integers \( n \in \mathbb{Z} \),
\[
\sum_{i=0}^{n} i = \frac{n(n + 1)}{2}.
\]
“Theorem 2”: For all natural numbers \( n \geq 1 \),
\[
\sum_{i=0}^{n} i = \frac{i(i + 1)}{2}.
\]
“Theorem 3”: For all natural numbers \( n \geq 1 \),
\[
\sum_{i=0}^{n} i = \frac{n(n - 1)}{2}.
\]
Expert Solution
Step 1: Flaw in theorem 1
We see, in integers, there are positive numbers and negative numbers both.
The given statement holds only for positive integers.
For example, if we take n to be negative, say n = -1.
Then, LHS evaluates to 0 + (-1) = -1
RHS evaluates to -1(-1+1)/2 = 0
So, LHS ≠ RHS.
Hence, in the statement, it should be
" For all non-nagative integers n"
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
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,
