Thread: 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

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 gives...



Quadratic equation - Wikipedia, the free encyclopedia

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

The general form of a quadratic equation is . a is the coefficient of , b is the coefficient of x, and c is the constant (or, if it's easier, imagine it as the coefficient of ).

alright thanks for all the help!