Answer to Question #197186 in Discrete Mathematics for sakibur rahman

Question #197186

Find the solution of the recurrence relation: xn=3xn-1 + 1, where,  x0=4


1
Expert's answer
2021-05-24T16:21:54-0400

Let us find the solution of the recurrence relation: "x_n=3x_{n-1} + 1", where "x_0=4."

For this firstly, let us solve the characteristic equation of the homogeneous equation "x_n-3x_{n-1}=0" :

"k-3=0" or "k=3." It follows that the particular solution of the equation is "x_n=a=const", where "a=3a+1," and hence "a=-\\frac{1}{2}." The solution of the recurrence relation: "x_n=3x_{n-1} + 1"

is "x_n=C\\cdot3^n-\\frac{1}{2}." Taking into account that "x_0=4," we conclude that "4=x_0=C-\\frac{1}{2}", and thus "C=\\frac{9}{2}." Consequently,

"x_n=\\frac{9}{2}\\cdot3^n-\\frac{1}{2}=\\frac{3^{n+2}-1}{2}."


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS