Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Solve the first-order linear recurrence relation: Sn+1 = Sn + 2, with S0=1.

Solve the first-order linear recurrence relation: Sn+1 = Sn + 2, with S0=1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Solve the first-order linear recurrence relation: Sn+1 = Sn + 2, with S0=1. You may use the general solution given on P.342.

Therefore, the general solution of the recurrence equation Sp+1 = aS, +c for Vn eN (8.2.1) is given in two parts: if a = 1, S, = 1+nc for Vn e N; if a + 1, S„ = a"A +• for Vn e N. 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, 1 = J – je. I/ since S; = I+jc // where So =1 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 = 1-, In fact, 1- a a particular solution is determined by a specific, numerical value J for any (particular) entry, S,. Solving the equation J = Ad + for A, a we get A = // But what if a = 0? // One particular “particular solution" has A = 0.
Transcribed Image Text:Therefore, the general solution of the recurrence equation Sp+1 = aS, +c for Vn eN (8.2.1) is given in two parts: if a = 1, S, = 1+nc for Vn e N; if a + 1, S„ = a"A +• for Vn e N. 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, 1 = J – je. I/ since S; = I+jc // where So =1 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 = 1-, In fact, 1- a a particular solution is determined by a specific, numerical value J for any (particular) entry, S,. Solving the equation J = Ad + for A, a we get A = // But what if a = 0? // One particular “particular solution" has A = 0.
8.2 Solving First-Order Linear Recurrence Equations A first-order linear recurrence equation relates consecutive entries in a sequence by an equation of the form Sn+1 = aS, +c for Vn in the domain of S. (8.2.1) But let's assume that the domain of S is N. Let's also assume that a + 0; otherwise, S, = c for Vn > 0, and the solutions to (8.2.1) are not very interesting. I/ What are they? We saw in Chap. 3, that when a = 1, any sequence satisfying (8.2.1) is an arithmetic sequence, and when S is defined on N and So is some initial value I, S, = 1+nc for Vn e N. // Theorem 3.6.4 Also, when e = 0, any sequence satisfying (8.2.1) is a geometric sequence, and when S is defined on N and So is some initial value I, S, = d"I for Vn e N. I/ Theorem 3.6.7 Furthermore, Theorem 3.6.8 gives a formula for the sum of the first (n + 1) terms of a geometric series when a + 1: a°1+a'I+a²I+.. +a'I = 1(1+a+a² +... +a") = 1ª а - 1
Transcribed Image Text:8.2 Solving First-Order Linear Recurrence Equations A first-order linear recurrence equation relates consecutive entries in a sequence by an equation of the form Sn+1 = aS, +c for Vn in the domain of S. (8.2.1) But let's assume that the domain of S is N. Let's also assume that a + 0; otherwise, S, = c for Vn > 0, and the solutions to (8.2.1) are not very interesting. I/ What are they? We saw in Chap. 3, that when a = 1, any sequence satisfying (8.2.1) is an arithmetic sequence, and when S is defined on N and So is some initial value I, S, = 1+nc for Vn e N. // Theorem 3.6.4 Also, when e = 0, any sequence satisfying (8.2.1) is a geometric sequence, and when S is defined on N and So is some initial value I, S, = d"I for Vn e N. I/ Theorem 3.6.7 Furthermore, Theorem 3.6.8 gives a formula for the sum of the first (n + 1) terms of a geometric series when a + 1: a°1+a'I+a²I+.. +a'I = 1(1+a+a² +... +a") = 1ª а - 1
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS