Answer to Question #302223 in Discrete Mathematics for salomo

Question #302223

Qn 3. Consider a sequence {an}∞

n=0 of real numbers defined as follows:

a0 = 3, a1 = 2, a2 = 12, and

2an+3 − an+2 − 8an+1 + 4an = 0 for all n ≥ 0.

(i) What is a99?

(ii) Find a formula for 2a2n+1 − a2n in terms of n


1
Expert's answer
2022-02-28T14:06:59-0500

The characteristic polynomial of this recurrence relation is


2r3r28r+4=02r^3-r^2-8r+4=0

(r2)(2r2+3r2)=0(r-2)(2r^2+3r-2)=0

(r2)(r+2)(2r1)=0(r-2)(r+2)(2r-1)=0

The characteristic roots are r1=2,r2=1/2,r3=2.r_1=-2, r_2=1/2, r_3=2.

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


an=α1(2)n+α2(12)n+α3(2)na_n=\alpha_1(-2)^n+\alpha_2(\dfrac{1}{2})^n+\alpha_3(2)^n

To find the constants α1,α2,α3,\alpha_1,\alpha_2,\alpha_3, use the initial conditions. This gives


a0=α1+α2+α3=3a_0=\alpha_1+\alpha_2+\alpha_3=3

a1=2α1+12α2+2α3=2a_1=-2\alpha_1+\dfrac{1}{2}\alpha_2+2\alpha_3=2

a2=4α1+14α2+4α3=12a_2=4\alpha_1+\dfrac{1}{4}\alpha_2+4\alpha_3=12

α1=1,α2=0,α3=2\alpha_1=1, \alpha_2=0, \alpha_3=2

Hence, the unique solution to this recurrence relation and the given initial

conditions is the sequence {an} with


an=(2)n+2(2)na_n=(-2)^n+2(2)^n

(i)


a99=(2)99+2(2)99=(2)99a_{99}=(-2)^{99}+2(2)^{99}=(2)^{99}

(ii)


2a2n+1a2n=2(2)2n+1+4(2)2n+12a_{2n+1}-a_{2n}=2(-2)^{2n+1}+4(2)^{2n+1}

((2)2n+2(2)2n)-((-2)^{2n}+2(2)^{2n})

=4(2)2n+8(2)2n(2)2n2(2)2n=-4(2)^{2n}+8(2)^{2n}-(2)^{2n}-2(2)^{2n}

=(2)2n=(4)n=(2)^{2n}=(4)^{n}

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