Solve the first-order linear recurrence relation: Sn+1 = 5 Sn + 1, with S0=1. You may use the general solution given on P.342.
Solve the first-order linear recurrence relation: Sn+1 = 5 Sn + 1, with S0=1. You may use the general solution given on P.342.
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
Related questions
Question
Solve the first-order linear recurrence relation: Sn+1 = 5 Sn + 1, with S0=1. You may use the general solution given on P.342.
![342
8 Sequences and Series
is given in two parts:
if a = 1,
for Vn E N;
for Vn e N.
Sn = I+nc
%3|
if a + 1,
Sn = a"A+
1
- a
When a = 1, any particular solution is obtained by determining a specific,
numerical value for I. In fact, a particular solution is determined by a specific,
numerical value J for any (particular) entry, S;. Solving the equation
J = I+jc for I,
I = J – jc.
// since S; = I+ jc
// where So =I
we get
// One particular “particular solution" has I = 0.
When a + 1, any particular solution is obtained by determining a specific,
numerical value for A; if the starting value I is given, then A = I -
In fact,
1 - a
a particular solution is determined by a specific, numerical value J for any
(particular) entry, S;. Solving the equation
J = Aa' +
for A,
- a
we get
A =
aJ
// But what if a = 0?
1 al
// One particular “particular solution" has A = 0.
Example 8.2.1: The Towers of Hanoi
The recurrence equation for the number of moves in the Towers of Hanoi
problem is a first-order linear recurrence equation:
T = 2T,-1+1.
1
2 and c =
= -1, and any sequence T that satisfies
2
Here a =
1, so
1
1
a
this RE is given by the formula
T, = 2"[1 – (–1)] +(-1)
= 2"[I + 1] – 1.
Assuming T has domain N and denoting To by I, we saw at the beginning of this
chapter several particular solutions:
= (0, 1,3, 7, 15, 31,...);
T = (2,5,11, 23, 47, 95, ...);
// T, = 2" [0+1] – 1 = 2"
if I = 0, then
if I = 2, then
if I = 4, then
if I = -1, then T = (-1,–1,-1,-1,–1,...). // T, = 2"(–1+1] – 1=
- 1.
// Tn = 2"[2+1] – 1 = 3 × 2" – 1.
T = (4,9, 19, 39, 79, 159, ...); // T, = 2" [4 + 1] – 1 = 5 x 2" – 1.
-1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4792360-22b4-40c0-a48a-02ce03c9f6f7%2Fc691280d-83f7-4552-be1b-ebaafb1f5d71%2F582etk_processed.png&w=3840&q=75)
Transcribed Image Text:342
8 Sequences and Series
is given in two parts:
if a = 1,
for Vn E N;
for Vn e N.
Sn = I+nc
%3|
if a + 1,
Sn = a"A+
1
- a
When a = 1, any particular solution is obtained by determining a specific,
numerical value for I. In fact, a particular solution is determined by a specific,
numerical value J for any (particular) entry, S;. Solving the equation
J = I+jc for I,
I = J – jc.
// since S; = I+ jc
// where So =I
we get
// One particular “particular solution" has I = 0.
When a + 1, any particular solution is obtained by determining a specific,
numerical value for A; if the starting value I is given, then A = I -
In fact,
1 - a
a particular solution is determined by a specific, numerical value J for any
(particular) entry, S;. Solving the equation
J = Aa' +
for A,
- a
we get
A =
aJ
// But what if a = 0?
1 al
// One particular “particular solution" has A = 0.
Example 8.2.1: The Towers of Hanoi
The recurrence equation for the number of moves in the Towers of Hanoi
problem is a first-order linear recurrence equation:
T = 2T,-1+1.
1
2 and c =
= -1, and any sequence T that satisfies
2
Here a =
1, so
1
1
a
this RE is given by the formula
T, = 2"[1 – (–1)] +(-1)
= 2"[I + 1] – 1.
Assuming T has domain N and denoting To by I, we saw at the beginning of this
chapter several particular solutions:
= (0, 1,3, 7, 15, 31,...);
T = (2,5,11, 23, 47, 95, ...);
// T, = 2" [0+1] – 1 = 2"
if I = 0, then
if I = 2, then
if I = 4, then
if I = -1, then T = (-1,–1,-1,-1,–1,...). // T, = 2"(–1+1] – 1=
- 1.
// Tn = 2"[2+1] – 1 = 3 × 2" – 1.
T = (4,9, 19, 39, 79, 159, ...); // T, = 2" [4 + 1] – 1 = 5 x 2" – 1.
-1.
Expert Solution
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
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Recommended textbooks for you
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education