# linear algebra

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. .

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1. 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

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.

https://math.stackexchange.com/questions/731916/strategies-for-developing-explicit-formulas-for-nth-term-given-recurrence-relati

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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

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3. nevermind, the pdf shows it very clearly

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