Answer to Question #271551 in Discrete Mathematics for nur

Question #271551

Suppose a recurrence relation




an=4an−1−4an−2



where a1=14 and a2=40




can be represented in explicit formula, either as:




Formula 1:



an=pxn+qnxn



or




Formula 2:



an=pxn+qyn






where



x



and



y



are roots of the characteristic equation.







Determine



p



and



q



.




1
Expert's answer
2021-11-26T13:43:46-0500
an4an1+4an2=0a_n-4a_{n−1}+4a_{n−2}=0

characteristic equation:


r24r+4=0r^2-4r+4=0

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

r=2r=2 is the only characteristic root. Therefore we know that the solution to the recurrence relation has the form


an=a2n+bn2na_n=a2^n+bn2^n

Given a1=14,a2=40a_1=14 ,a_2=40


a1=a(21)+b(1)(21)=14a_1=a(2^1)+b(1)(2^1)=14

a2=a(22)+b(2)(22)=40a_2=a(2^2)+b(2)(2^2)=40

a+b=7a+b=7a+2b=10\\ a+2b=10


a=4a=4b=3b=3

Therefore the solution to the recurrence relation is


an=42n+3n2na_n=4\cdot2^n+3n\cdot2^n

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