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Math Help - Recursion Characteristic Equation

  1. #1
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    Recursion Characteristic Equation

    Alice wants to stack plastic cups of different colors together. The plastic cups come in 4 colors - red,green,blue and white. Let a n be the number of ways to stack n of these plastic cups so that there are no consecutive white plastic cups. Find a recurrence relation for a n and solve the recurrence relation.

    Model solution:
    a n = 3*a n-1 + 3*a n-2
    a1=4, a2=15
    x^2 -3x -3 = 0
    X1 = (3+ √21)/2
    X2 = (3- √21)/2
    .
    .
    .
    What i want to ask is where did he get this equation:(3+ √21)/2 come from? i know it has something got to do with characteristic equation, but i cant figure out where did the 3 , √21 and /2 come from? Please help
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  2. #2
    Super Member Deadstar's Avatar
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    Quote Originally Posted by hugo84 View Post
    Alice wants to stack plastic cups of different colors together. The plastic cups come in 4 colors - red,green,blue and white. Let a n be the number of ways to stack n of these plastic cups so that there are no consecutive white plastic cups. Find a recurrence relation for a n and solve the recurrence relation.

    Model solution:
    a n = 3*a n-1 + 3*a n-2
    a1=4, a2=15
    x^2 -3x -3 = 0
    X1 = (3+ √21)/2
    X2 = (3- √21)/2
    .
    .
    .
    What i want to ask is where did he get this equation3+ √21)/2 come from? i know it has something got to do with characteristic equation, but i cant figure out where did the 3 , √21 and /2 come from? Please help
    Solving x^2 - 3x - 3 = 0 gives...

    \frac{-(-3) \pm \sqrt{(-3)^2 - 4\cdot1\cdot(-3)}}{2} = \frac{3 \pm \sqrt{21}}{2}

    Quadratic equation - Wikipedia, the free encyclopedia
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  3. #3
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    How do you determine what is a,b and c from the equation
    x^2 -3x-3=0
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  4. #4
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    The general form of a quadratic equation is ax^2+bx+c=0. a is the coefficient of x^2, b is the coefficient of x, and c is the constant (or, if it's easier, imagine it as the coefficient of x^0).
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  5. #5
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    alright thanks for all the help!
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