Subject: 7.1 - number 5 From: "William J. Rapaport" Date: Tue, 15 Dec 2009 12:45:05 -0500 (EST) A student writes, concerning the problem identified in the subject line of this message: "I'm a little confused by this question. Part a says it is a recurrence relation because 0 = 0 but when an = 0, doesn't an-1= -1 and an-2 = -2 which means whenyou plug them in you get: 0 = 8(-1) - 16(-2) 0 = -8 + 32 0 = 24 I guess you are supposed to put 0 in for an an-1 and an-2 but that doesn't make sense." Reply: The question states: Is the sequence {a_n} a solution of the recurrence relation a_n=8a_n-1 - 16a_n-2 if a_n = 0? So, it doesn't "say it is a recurrence relation because 0=0". What is the sequence {a_n} if a_n=0? It is: 0,0,0,0,... So, a1=0, a2=0, and therefore a3=8a2 - 16a1 = 8*0 - 16*0 = 0. Similarly, a4=0, a5=0, and, in general a_n=8a_n-1 - 16a_n-2 = 0. Therefore, a_n = 0 is indeed a solution of that recurrence relation. I don't see why you think that a_n-1 = -1 if a_n = 0. If a_n = 0 for all n, then a_n-1 = 0. I don't quite understand why you think "that doesn't make sense". ======================================================================== Subject: Re: 7.1 - number 5 From: "William J. Rapaport" Date: Tue, 15 Dec 2009 12:57:20 -0500 (EST) The student responds: "I was thinking, a_n = 0, and that a_n-1 was the number before 0 which is -1 and a_n-2 is the number before a_n-1 which would be -2." Reply: a_n-1 is the term before a_n. To compute its value, you need to know a formula for a_n. The problem gives you two different formulas, one recursive, and one explicit. You need to decide if they are the same for all values of n.