Use generating functions to solve the following recurrence r

Use generating functions to solve the following recurrence relations: A_n - 3a_n-1 = n, greaterthenorequalto 1, a_0 =1 a_n - a_n-1 = 3n^2 + 5 n^3, n greaterthenorequalto 1, a_0 = 1

Solution

(a) given that a_n-3a_n-1=n n>1,a0=1

here the general equation for this is x^2-3x+3=0

by solving this we got the values x=3 or -3

here we have an=A(x)^n

an=A(2)^1

a0=A(2) here x^n+1=2x^n

by dividing with x^n wew get x=2

1=A(2)

then A=1/2