Asked by Julius
Suppose that the sequence x0, x1, x2... is defined by x0 = 1, x1 = 4, and xk+2 = −6xk+1−8xk for k≥0. Find a general formula for xk. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. .
No idea how to go about this, please help
No idea how to go about this, please help
Answers
Answered by
Steve
x0 = 1
x1 = 4
x2 = -6*4 - 8*1 = -32
x3 = -6(-32)-8*4 = 160
x4 = -6(160)-8(-32) = -704
x5 = -6(-704)-8(160) = 2944
Read about characteristic functions for recursive sequences. For example, at
www.csee.umbc.edu/~stephens/203/PDF/8-3.pdf
It shows that the characteristic equation for this sequence is
x^2 = -6x-8
x^2+6x+8 = 0
(x+4)(x+2) = 0
x = -4,-2
So, the general formula for the sequence is
xn = C(-4)^n+D(-2)^n
plugging x0 and x1, we get
xn = -3(-4)^n + 4(-2)^n
You can see that it works for the terms given above.
This article is also useful.
https://math.stackexchange.com/questions/731916/strategies-for-developing-explicit-formulas-for-nth-term-given-recurrence-relati
remember: google is your friend!
x1 = 4
x2 = -6*4 - 8*1 = -32
x3 = -6(-32)-8*4 = 160
x4 = -6(160)-8(-32) = -704
x5 = -6(-704)-8(160) = 2944
Read about characteristic functions for recursive sequences. For example, at
www.csee.umbc.edu/~stephens/203/PDF/8-3.pdf
It shows that the characteristic equation for this sequence is
x^2 = -6x-8
x^2+6x+8 = 0
(x+4)(x+2) = 0
x = -4,-2
So, the general formula for the sequence is
xn = C(-4)^n+D(-2)^n
plugging x0 and x1, we get
xn = -3(-4)^n + 4(-2)^n
You can see that it works for the terms given above.
This article is also useful.
https://math.stackexchange.com/questions/731916/strategies-for-developing-explicit-formulas-for-nth-term-given-recurrence-relati
remember: google is your friend!
Answered by
Julius
If I may ask a follow up question:
everything makes perfect sense up until the last line. How exactly did you come up with -3 and 4? I've been trying to solve similar problems and the coefficient seems the only thing off
everything makes perfect sense up until the last line. How exactly did you come up with -3 and 4? I've been trying to solve similar problems and the coefficient seems the only thing off
Answered by
Julius
nevermind, the pdf shows it very clearly
Answered by
Hera
I am so lost, how did you get -3 and +4? Also, the PDF is not opening for me, so I'm unable to see what you have shared.