# Thread: Recursion Characteristic Equation

1. ## 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

2. Originally Posted by hugo84
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

3. How do you determine what is a,b and c from the equation
x^2 -3x-3=0

4. 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 $x^0$).

5. alright thanks for all the help!