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".
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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.