Answer to Question #160838 in Discrete Mathematics for eyasu

Question #160838

solve these non-homogeneous recurrence relation an+ 4an-1 − 5an-2=n+2 where a0=1 & a1=-1

1
Expert's answer
2021-02-04T07:37:09-0500

A solution "b_n" to the non-homogeneous recurrence is similar to f(n).

Guess:

"b_n=cn+d"

Then:

"cn+d+4(c(n-1)+d)-5(c(n-2)+d)=n+2"

"n-4=0"

"c=d=b_n=0"


Solution of the given recurrence is

"a_n=b_n+h_n"

where "h_n" is a solution for the associated homogeneous recurrence:

"h_n+4h_{n-1}-5h_{n-2}=0"

In our case: "a_n=h_n"


Characteristic equation:

"r^2+4r-5=0"

"r_1=\\frac{-4-\\sqrt{16+20}}{2}=-5, r_2=1"

"a_n=\\alpha_1r_1^n+\\alpha_2r_2^n"

"a_n=\\alpha_1(-5)^n+\\alpha_2"


"a_0=\\alpha_1+\\alpha_2=1"

"a_1=-5\\alpha_1+\\alpha_2=-1"

"-5\\alpha_1+(1-\\alpha_1)=-1"

"\\alpha_1=1\/3,\\alpha_2=2\/3"


Solution:

"a_n=\\frac{1}{3}(-5)^n+\\frac{2}{3}"


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