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. .

No idea how to go about this, please help

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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

    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!

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  2. 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

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  3. nevermind, the pdf shows it very clearly

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