Neap SmartStudy Guide Help

[sam.utute]

Hey guys,

I'm stuck on a question in the Neap book (Practice Exam 3, Module 1, Question 9 if anyone else has the book [pg148]), and despite availing my Further teacher for help, he has been unable to clear up the issue.

The Quesiton:

A graph is produced or a particular difference equation.

As I don't have a scanner, here's the written version of the graph (imagination required):
{3, 1, -2, -3, -1, 2, 3, 1, -2, -3, -1, 2, etc. etc.} repeated n times.

The difference equation itself:
A. must be first-order
B. could be first or second-order
C. could be second-order but not first-order
D. must be second-order
E. is a Fibonacci sequence.

After eliminating the obivious wrong answer (E, Fibonacci), I still ended up guessing the answer, and got it wrong. I have no idea how to work this out, or what piece of theory it relates to.

Answer: B  :idiot2:
Options A and E are the most easily discounted options. This is clearly not the graph of the Fibonacci sequence which is always increasing, nor must it be first-order. A second-order equation could produce this pattern. For every first-order difference equation, every incidence of a certain sequence value must be followed by the same value [umm... what!?  :uglystupid2:]. This is also true for the sequence concerned. It could be either first or second-order.

Not only do I NOT understand the overtly complex answer, I can't even make sense of what its saying (i.e. I DON'T GET IT). 

Please, please, Maths pros out there, HELP!

Counting on VCE Notes!
Sam 

[Martoman]

Instantly the thing that jumps out at me is:

, meaning, the term after is generated by subtracting the previous term from its previous term. So is created from 1-3 = -2 and similarly -2-1 = -3 = and so on.

Thus we are relating two steps in this equation, and so it can be described by a second order difference equation.

I cannot understand how it can be described by a first order difference equation. Using algebra and saying gives the general form of a first order difference equation. Trying to model this to any one of the sequence say and also

Magic simeltaneous equations gives:




subtracting to eliminate that dastardly "d"





and thus d = 7 so

which doesn't hold true for a number of cases listed, = 1*-2 +7 = 5 this doesn't equal -2 so we have a problem. I don't think a first order equation can be thus modelled on this sequence.

I'd go with D
 

[sam.utute]

Thank you Martoman ! I get the second-order equation now.

My assumption for first-order:

I'm not sure that this is correct, but I came up with a first-order difference equation.

t n+3 = -1 x t n, where t1=3, t2=1, t3=-2

So, t4:

t4 = -1 x t1
    = -1 x 3
    = -3

This equation works for the whole sequence. I'm just not sure that with first-order equations you are allowed to specify the first three values (i.e. 3, 1, -2). Without that the equation would not work.

[Martoman]

Thats a second order diff equation... its jumps two. A first order relates the next term with the previous.

[sam.utute]

Hmmmmmm....

Then how did Neap get the answer as B? o.O

[Martoman]

Most likely an error.