The inductive step of an inductive proof shows that for k > 0, if E 2' = 2k+l – 1 then E 2 = 2*+2 j=0 In which step of the proof is the inductive hypothesis used? E 2 3(2*+1 – 1) + 2*+1 (Step 2) E, 2 + 2*+1 (Step 1) j=0 %3D = 2· 2k+1 – 1 (Step 3) = 2k+2 - 1 (Step 4) O Step 3 O Step 2 O Step 1 O Step 4
The inductive step of an inductive proof shows that for k > 0, if E 2' = 2k+l – 1 then E 2 = 2*+2 j=0 In which step of the proof is the inductive hypothesis used? E 2 3(2*+1 – 1) + 2*+1 (Step 2) E, 2 + 2*+1 (Step 1) j=0 %3D = 2· 2k+1 – 1 (Step 3) = 2k+2 - 1 (Step 4) O Step 3 O Step 2 O Step 1 O Step 4
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
![The inductive step of an inductive proof shows that for \( k \geq 0 \), if \( \sum_{j=0}^{k} 2^j = 2^{k+1} - 1 \) then \( \sum_{j=0}^{k+1} 2^j = 2^{k+2} - 1 \).
In which step of the proof is the inductive hypothesis used?
\[
\begin{align*}
\sum_{j=0}^{k+1} 2^j & = \sum_{j=0}^{k} 2^j + 2^{k+1} \quad \text{(Step 1)} \\
& = \left(2^{k+1} - 1\right) + 2^{k+1} \quad \text{(Step 2)} \\
& = 2 \cdot 2^{k+1} - 1 \quad \text{(Step 3)} \\
& = 2^{k+2} - 1 \quad \text{(Step 4)}
\end{align*}
\]
Choose one:
- Step 3
- Step 2
- Step 1
- Step 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58591baa-c0bc-433b-b6a8-54571afeaf18%2F4b6730a8-a537-45ef-bdb7-95fa712a5ba4%2Fq2v4no8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The inductive step of an inductive proof shows that for \( k \geq 0 \), if \( \sum_{j=0}^{k} 2^j = 2^{k+1} - 1 \) then \( \sum_{j=0}^{k+1} 2^j = 2^{k+2} - 1 \).
In which step of the proof is the inductive hypothesis used?
\[
\begin{align*}
\sum_{j=0}^{k+1} 2^j & = \sum_{j=0}^{k} 2^j + 2^{k+1} \quad \text{(Step 1)} \\
& = \left(2^{k+1} - 1\right) + 2^{k+1} \quad \text{(Step 2)} \\
& = 2 \cdot 2^{k+1} - 1 \quad \text{(Step 3)} \\
& = 2^{k+2} - 1 \quad \text{(Step 4)}
\end{align*}
\]
Choose one:
- Step 3
- Step 2
- Step 1
- Step 4
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 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,
