Question

Let a_0=0 a_1=2 and a_2=5. Use generating function to solve the recurrence equation a_(n+3) = 5a_(n+2)-7a_(n+1)+3a_(n)+2^n for n>=0

Answer 1

\(a_{n+3}= 5a_{n+2}-7a_{n+1}+3a_n+2^n\)
\(a_{n+3}-5a_{n+2}+7a_{n+1}-3a_n=2^n\)
characteristic equation for homogeneous part is
x^3 -5x^2+7x-3 =0
which gives x = 3 and x =1 double root
so, \(a_n = A*3^n+ B +Cn

Answer 2

got me so far?

Answer 3

you have to solve for nonhomogeneous part, It is awhile I didn't solve this stuff. Hopefully it makes sense to you.