Consider the following recurrence relation: if n = 0 if n > 0. C(n) = - {1+3.0 (n+3·C(n-1) Prove by induction that C(n) = 3n+1-2n - 3 for all n ≥ 0. 4 3n+1 - 2n (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 3-1+1-2(k-1)-3 4 3²+1 -3 4 -6k-3 4 X 4 3+1-2k-3 by inductive hypothesis X 0+1 for some k > 0. - 2.0 0 ✓ , so they match.
Consider the following recurrence relation: if n = 0 if n > 0. C(n) = - {1+3.0 (n+3·C(n-1) Prove by induction that C(n) = 3n+1-2n - 3 for all n ≥ 0. 4 3n+1 - 2n (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 3-1+1-2(k-1)-3 4 3²+1 -3 4 -6k-3 4 X 4 3+1-2k-3 by inductive hypothesis X 0+1 for some k > 0. - 2.0 0 ✓ , so they match.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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need help with the 2 boxes with x mark
Transcribed Image Text:Consider the following recurrence relation:
=
C(n) =
) = {0+3. C(n-1) ifn>0.
Prove by induction that C(n) = 3+1 -2n-3 for all n ≥ 0.
4
(Induction on n.) Let f(n) = 3n+1 - 2n-
Base Case: If n = 0, the recurrence relation says that C(0) = 0, and the formula says that f(0) =
=k+ 3.1
Inductive Hypothesis: Suppose as inductive hypothesis that C(k-1)=f(k-1)
AK
Inductive Step: Using the recurrence relation,
C(k)=k+ 3 C(k-1), by the second part of the recurrence relation
-3
3+1-6k-3
4
3−1+1 -2(k-1) – 3
4
4
3k+1-2k-3
4
X
by inductive hypothesis
X
0+1
for some k > 0.
4
-2.0-3
0
✓
, so they match.
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