Answer to Question #268859 in Discrete Mathematics for Zobird

Question #268859

4. (2 points) Solve these recurrence relations together with the initial conditions given. Each is worth 1 point.

a) an = 6an-1 - 11an-3 +6an-3 for n > 3, a0 = 4, a1 = 6, a2 = 12

b) an = an–1 +9an-2 – 9an-3 for n > 3, a0 = 6, a1= 0, a2 = 30


1
Expert's answer
2021-11-22T10:49:31-0500

a)

The characteristic polynomial of this recurrence relation is


r36r2+11r6=0r^3-6r^2+11r-6=0

r2(r1)5r(r1)+6(r1)=0r^2(r-1)-5r(r-1)+6(r-1)=0

(r1)(r25r+6)=0(r-1)(r^2-5r+6)=0

(r1)(r2)(r3)=0(r-1)(r-2)(r-3)=0

r1=1,r2=2,r3=3r_1=1, r_2=2, r_3=3

Hence, the solutions to this recurrence relation are of the form


an=α11n+α22n+α33na_n=\alpha_1\cdot1^n+\alpha_2\cdot2^n+\alpha_3\cdot3^n

Use the initial conditions


a0=4=α1+α2+α3a_0=4=\alpha_1+\alpha_2+\alpha_3a1=6=α1+2α2+3α3a_1=6=\alpha_1+2\alpha_2+3\alpha_3a2=12=α1+4α2+9α3a_2=12=\alpha_1+4\alpha_2+9\alpha_3

α1+α2+α3=4\alpha_1+\alpha_2+\alpha_3=4α2+2α3=2\alpha_2+2\alpha_3=22α2+6α3=62\alpha_2+6\alpha_3=6

α1+α2+α3=4\alpha_1+\alpha_2+\alpha_3=4α2+2α3=2\alpha_2+2\alpha_3=2α2+3α3=3\alpha_2+3\alpha_3=3


α1=3\alpha_1=3α2=0\alpha_2=0α3=1\alpha_3=1

Hence, the unique solution to this recurrence relation and the given initial conditions is the sequence {an}\{a_n\} with


an=3+3na_n=3+3^n

b)

The characteristic polynomial of this recurrence relation is


r3r29r+9=0r^3-r^2-9r+9=0

r2(r1)9(r1)=0r^2(r-1)-9(r-1)=0

(r1)(r3)(r+3)=0(r-1)(r-3)(r+3)=0

r1=1,r2=3,r3=3r_1=1, r_2=-3, r_3=3

Hence, the solutions to this recurrence relation are of the form


an=α11n+α2(3)n+α33na_n=\alpha_1\cdot1^n+\alpha_2\cdot(-3)^n+\alpha_3\cdot3^n

Use the initial conditions


a0=6=α1+α2+α3a_0=6=\alpha_1+\alpha_2+\alpha_3a1=0=α13α2+3α3a_1=0=\alpha_1-3\alpha_2+3\alpha_3a2=30=α1+9α2+9α3a_2=30=\alpha_1+9\alpha_2+9\alpha_3

α1+α2+α3=6\alpha_1+\alpha_2+\alpha_3=64α22α3=64\alpha_2-2\alpha_3=68α2+8α3=248\alpha_2+8\alpha_3=24

α1+α2+α3=6\alpha_1+\alpha_2+\alpha_3=62α2α3=32\alpha_2-\alpha_3=3α2+α3=3\alpha_2+\alpha_3=3


α1=3\alpha_1=3α2=2\alpha_2=2α3=1\alpha_3=1

Hence, the unique solution to this recurrence relation and the given initial conditions is the sequence {an}\{a_n\} with


an=3+2(3)n+3na_n=3+2\cdot(-3)^n+3^n


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