My Math Forum Discrete Math Help?

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April 17th, 2013, 10:02 AM   #1
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Discrete Math Help?

Not sure if this is the right board, but here goes:

[attachment=0:2phg64no]Logic.png[/attachment:2phg64no]

Can someone tell me how to do these two problems or direct me to the right board to post for it? I have no idea where to start because I thought a Fibonacci Sequence is adding numbers that are right next to each other to find the next number of the sequence. So I thought if the first number was 2, wouldn't the sequence be 2,2,4,6,10? Why does n=4 turn out to be 3?

And the recursion one I have no idea what to do either.
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April 17th, 2013, 11:41 AM   #2
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Re: Discrete Math Help?

Quote:
 Originally Posted by pjlloyd100 Can someone tell me how to do these two problems or direct me to the right board to post for it? I have no idea where to start because I thought a Fibonacci Sequence is adding numbers that are right next to each other to find the next number of the sequence. So I thought if the first number was 2, wouldn't the sequence be 2,2,4,6,10? Why does n=4 turn out to be 3?
The sequence starts 2 and then some other number, call it x. The second term is x+2. The third term is x + x+2 = 2x+2. The fourth term is x+2 + 2x+2 = 3x+4 = 4. But you also know the fourth term is 4.

 April 17th, 2013, 11:53 AM #3 Member   Joined: Oct 2012 Posts: 39 Thanks: 0 Re: Discrete Math Help? What how is the first number x when it states that it is 2? And how is the last number 4 when it states that it is 3? You totally lost me. Also, how would I find a sequence from the information you just told me?..
 April 17th, 2013, 11:54 AM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,193 Thanks: 504 Math Focus: Calculus/ODEs Re: Discrete Math Help? 1.) Yes, using the recursive definition, you obtain: $2+X_1=X_2$ $X_1+X_2=X_3$ $X_2+X_3=3$ You have a linear 3X3 system, which you can easily solve 2.) The associated characteristic equation is: $r^2\,-\,7r\,+\,10\,=\,0$ Find the roots, call them $r_1,r_2$, and the closed form is: $W_n=k_1r_1^n+k_2r_2^n$ Now you may determine the parameters $k_i$ from the given initial values. Post what you find, and we can offer further guidance if you get stuck.
 April 17th, 2013, 12:07 PM #5 Member   Joined: Oct 2012 Posts: 39 Thanks: 0 Re: Discrete Math Help? Sigh.
April 17th, 2013, 12:22 PM   #6
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Re: Discrete Math Help?

Quote:
 Originally Posted by pjlloyd100 Sigh.
Does none of what I posted help you? Let's concentrate on the first problem. We are told to use:

$X_{n+2}=X_{n}+X_{n+1}$

where:

$X_0=2,\,X_4=3$

So, using the recursive definition, we know, with $n=0$, we have:

$X_{2}=X_{0}+X_{1}=2+X_{1}$ so we may arrange this as:

$2+X_1=X_2$

And the other two equations I obtained similarly for $n=1,\,2$

This gives you 3 equations in 3 unknowns, which are solvable by substitution/elimination. Does this make more sense?

 April 17th, 2013, 12:36 PM #7 Member   Joined: Oct 2012 Posts: 39 Thanks: 0 Re: Discrete Math Help? Ok. After Substitution/Elimination I got: Xo = 2 X1 = -1/3 X2 = 5/3 X3 = 4/3 X4 = 9/3 = 3 Am I done? Is that what is mean by find the sequence? I'm guessing I'm done because I do have a sequence of numbers 2, -1/3, 5/3, 4/3, 9/3
 April 17th, 2013, 12:41 PM #8 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,193 Thanks: 504 Math Focus: Calculus/ODEs Re: Discrete Math Help? Yes, all you need to define the sequence is two consecutive terms, as all the others may be found via the recursive definition. You have correctly solved the system, so you may state the sequence is: $X_{n+2}=X_{n}+X_{n+1}$ where: $X_0=2,\,X_1=-\frac{1}{3}$ Good work! Now, for the second problem, do you understand how the characteristic roots determine the closed form?
 April 17th, 2013, 01:07 PM #9 Member   Joined: Oct 2012 Posts: 39 Thanks: 0 Re: Discrete Math Help? Not really. I'm really confused on that problem.
 April 17th, 2013, 01:16 PM #10 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,193 Thanks: 504 Math Focus: Calculus/ODEs Re: Discrete Math Help? What method have you been taught to find the closed form of a linear homogeneous recursion?

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