Consider the following recurrence relation: -{+ C(n) = if n = 0 n+ 3. C(n-1) if n > 0. Prove by induction that C(n) = 3+1 -2n-3 for all n ≥ 0. 4 30+1 (Induction on n.) Let f(n) = 2n-1 4 Base Case: If n = 0, the recurrence relation says that C(0) = 0, and the formula says that f(0) = Inductive Hypothesis: Suppose as inductive hypothesis that C(k-1)=f(k-1) Inductive Step: Using the recurrence relation, C(K) = k + 3 = k + 3 X C(k-1), by the second part of the recurrence relation 3k-1+12(k-1) - 3 by inductive hypothesis X 0+1 for some k > 0. -2.0-3 , so they match.
Consider the following recurrence relation: -{+ C(n) = if n = 0 n+ 3. C(n-1) if n > 0. Prove by induction that C(n) = 3+1 -2n-3 for all n ≥ 0. 4 30+1 (Induction on n.) Let f(n) = 2n-1 4 Base Case: If n = 0, the recurrence relation says that C(0) = 0, and the formula says that f(0) = Inductive Hypothesis: Suppose as inductive hypothesis that C(k-1)=f(k-1) Inductive Step: Using the recurrence relation, C(K) = k + 3 = k + 3 X C(k-1), by the second part of the recurrence relation 3k-1+12(k-1) - 3 by inductive hypothesis X 0+1 for some k > 0. -2.0-3 , so they match.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 56EQ
Related questions
Question
100%
Need help with induction on N, and inductive hypothesis, they are incorrect.
![Consider the following recurrence relation:
-{+
C(n) =
if n = 0
n+ 3. C(n-1) if n > 0.
Prove by induction that C(n) = 3+1 -2n-3 for all n ≥ 0.
4
30+1
(Induction on n.) Let f(n) =
Base Case: If n = 0, the recurrence relation says that C(0) = 0, and the formula says that f(0) =
Inductive Hypothesis: Suppose as inductive hypothesis that C(k-1) = f(k-1)
=k+3.
Inductive Step: Using the recurrence relation,
C(K) = k + 3. C(k-1), by the second part of the recurrence relation
= 4k +
3-1+1.
3k+1
3k+1
4
2n-1
4
4
2k-3
- 6k-3
-2(k-1)-3
4
X
so, by induction, C(n) = f(n) for all n ≥ 0.
by inductive hypothesis
X
0+1
for some k > 0.
-2.0-3
, so they match.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15dc7053-7424-4cd9-bc21-341d9a11b287%2F3db45add-5fb6-479e-9d66-7c6a138c208c%2F1gv5s44_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following recurrence relation:
-{+
C(n) =
if n = 0
n+ 3. C(n-1) if n > 0.
Prove by induction that C(n) = 3+1 -2n-3 for all n ≥ 0.
4
30+1
(Induction on n.) Let f(n) =
Base Case: If n = 0, the recurrence relation says that C(0) = 0, and the formula says that f(0) =
Inductive Hypothesis: Suppose as inductive hypothesis that C(k-1) = f(k-1)
=k+3.
Inductive Step: Using the recurrence relation,
C(K) = k + 3. C(k-1), by the second part of the recurrence relation
= 4k +
3-1+1.
3k+1
3k+1
4
2n-1
4
4
2k-3
- 6k-3
-2(k-1)-3
4
X
so, by induction, C(n) = f(n) for all n ≥ 0.
by inductive hypothesis
X
0+1
for some k > 0.
-2.0-3
, so they match.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage