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Math Help - [SOLVED] Need help on this proof

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    Question [SOLVED] Need help on this proof

    Prove that a[n]=x*2^n+y*3^n is a solution to the equation
    a[n] = 5*a[n-1] - 6*a[n-2] when n>=2 ,x and y are integers and numbers in [] s are subscripts and ^ denotes power .
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  2. #2
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    Quote Originally Posted by gigglie View Post
    Prove that a[n]=x*2^n+y*3^n is a solution to the equation
    a[n] = 5*a[n-1] - 6*a[n-2] when n>=2 ,x and y are integers and numbers in [] s are subscripts and ^ denotes power .


    To prove a_n = x2^n+y3^n is a solution, substitute this in  a_n = 5a_{n-1}- 6a_{n-2} and see if it satisfies.

    a_n = 5a_{n-1}- 6a_{n-2} \Rightarrow 5(x2^{n-1}+y3^{n-1}) - 6(x2^{n-2}+y3^{n-2})

    = (10x - 6x)2^{n-2}+(15y - 6y)3^{n-2} = (4x)2^{n-2}+(9y)3^{n-2} = a_n


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~
    The standard way to solve these homogenus recurrences is to substitute a_n = r^n and see what value of r you get.Then the general solution is a linear combination of the obtained r^n

     a_n = 5a_{n-1}- 6a_{n-2}
    r^n = 5r^{n-1}- 6r^{n-2}
    r^2 = 5r- 6 \Rightarrow r = 2,3

    Thus a_n = x2^n+y3^n
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