Kenneth Rosen Edition 7 Exercise 8.2 Question 14 (Page No. 525)

Question

Find the solution to $a_{n} = 5a_{n−2}− 4a_{n−4} \:\text{with}\: a_{0} = 3, a_{1} = 2, a_{2} = 6, \:\text{and}\: a_{3} = 8.$

Answer 1

EQUATION

$r^{4}-5r^{2}+4=0$

$(r-1)(r-2)(r+1)(r+2)=0$

$r=1,-1,2,-2$

solution of recurence rquation

$a_n=A(1)^{n}+B(-1)^{n}+C(2)^{n}+D(-2)^{n}$

put value of n=0,1,2,3

3=A+B+C+D

2=A-B+2C-2D

6=A+B+4C+4D

8=A-B+8C-8D

VALUE OF A=1,B=,1C=1,D=0

$a_n=(1)^{n}+(-1)^{n}+(2)^{n}$