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Math Help - Recurrence Relation - Next Step

  1. #1
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    Recurrence Relation - Next Step

    The question is: An = An-1 + 6An-2 for n>=2, a0=3, a1=6

    So far I have...

    r^2-r-6=0 so the roots are r=-2,3.

    I am stuck here and not sure where to go. Can someone please help me! Thanks in advance.
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  2. #2
    Eater of Worlds
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    You have the quadratic we need to work with OK.

    Since the roots are -2 and 3, we have

    A\cdot (-2)^{n}+B\cdot 3^{n}

    Now, use the initial conditions, a_{0}=3, \;\ a_{1}=6

    If n=0, we have A\cdot (-2)^{0}+B\cdot 3^{0}=A+B=3

    If n=1, we have A\cdot (-2)^{1}+B\cdot 3^{1}=-2A+3B=6

    The two equations to solve are:

    A+B=3
    -2A+3B=6

    A=\frac{3}{5}, \;\ B=\frac{12}{5}

    \boxed{\frac{3}{5}\cdot (-2)^{n}+\frac{12}{5}\cdot 3^{n}}
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