Answer to Question #275326 in Discrete Mathematics for feey

Question #275326

Suppose a recurrence relation


an=an−1+20an−2

where a1=9 and a2=189


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.


**If the explicit formula is in the form of Formula 2, consider p > q.


 


Determine p and q

Answer:


1
Expert's answer
2021-12-06T16:27:07-0500

Let us solve the characteristic equation of the recurrence relation an=an120an2,a_n=a_{n−1}−20a_{n−2}, which is equivalent to anan120an2=0.a_n-a_{n−1}-20a_{n−2}=0. 

It follows that the characteristic equation x2x20=0x^2-x-20=0 is equivalent to (x+4)(x5)=0,(x+4)(x-5)=0, and hence has the roots x1=4,x2=5.x_1=-4, x_2=5. 

It follows that the solution of the recurrence equation is 

an=p(4)n+q(5)n.a_n=p\cdot (-4)^n+q\cdot (5)^n.

 Since a1=9a_1=9 and a2=189,a_2=189, we conclude that


9=a1=4p+5q9=a_1=-4p+5q189=a2=16p+25q189=a_2=16p+25q


9=4p+5q9=-4p+5q36+189=16p+16p+20q+25q36+189=-16p+16p+20q+25q


9=4p+5q9=-4p+5q225=45q225=45q

Therefore, q=5q=5 and p=4.p=4.



an=4(4)n+5(5)n.a_n=4(-4)^n+5(5)^n.


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