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A sequence (u,),nzo is defined recursively, with u, and u, given by 2un+2 – Un+1 – Un = 0 Two other sequences vn = Un+1 - Un and wn = 2un+1 + Un are defined Vn EN. a) Show that (vn)nzo is a geometric sequence b) Express vn in terms of n, u, and U1. c) Show that (wn)n20 is a constant sequence. Hence express w, in terms of u, and u..

A sequence (u,),nzo is defined recursively, with u, and u, given by 2un+2 – Un+1 – Un = 0 Two other sequences vn = Un+1 - Un and wn = 2un+1 + Un are defined Vn EN. a) Show that (vn)nzo is a geometric sequence b) Express vn in terms of n, u, and U1. c) Show that (wn)n20 is a constant sequence. Hence express w, in terms of u, and u..

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
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Solve Q2a, b & c explaining detailly each step

2. A sequence (un)nzo is defined recursively, with u, and u̟ given by 2un+2 – Un+1 – Un = 0 Two other sequences vn = Un+1 – Un and wn = 2un+1 + Un are defined Vn EN. a) Show that (vn)nzo is a geometric sequence b) Express vn in terms of n, u, and U1. c) Show that (w,n)nzo is a constant sequence. Hence express w, in terms of u, and u,. d) By solving wn – 2vn in two different ways, give an expression for un in terms of n, u, and uq. e) Consider the sum S, defined as n Uk k=0 Express Sn in terms of n, u, and u1. f) For what values of uo and uz does the sequence (Sn)n20 admits a finite limit and in that case express this limit in terms of u,.
Transcribed Image Text:2. A sequence (un)nzo is defined recursively, with u, and u̟ given by 2un+2 – Un+1 – Un = 0 Two other sequences vn = Un+1 – Un and wn = 2un+1 + Un are defined Vn EN. a) Show that (vn)nzo is a geometric sequence b) Express vn in terms of n, u, and U1. c) Show that (w,n)nzo is a constant sequence. Hence express w, in terms of u, and u,. d) By solving wn – 2vn in two different ways, give an expression for un in terms of n, u, and uq. e) Consider the sum S, defined as n Uk k=0 Express Sn in terms of n, u, and u1. f) For what values of uo and uz does the sequence (Sn)n20 admits a finite limit and in that case express this limit in terms of u,.
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