Help with recursive sequences SolutionI am going to slove 1s

Help with recursive sequences !

I am going to slove 1st question , the remaining follow same procedure

X(n+3) = 9X(n+2) -27X(n+1) +27X(n)

let X(n) = x^n , then the given equation becomes

x^(n+3) = 9x^(n+2) -27x^(n+1) + 27x^n

x^n.x^3 = x^n (9x^2 -27x +27)

x^3 = (9x^2 -27x +27)

x^3 -9x^2 +27x -27=0

clearly we can that one root is 3

so x^3 -9x^2 +27x -27=0 can be written as (x-3) (x^2 -6x +9) =0

so the roots are same 3,3,3

so general solution will be X(n) = a(3)^n + nb(3)^n + cn^2(3)^n

X(1) = a(3) + 3b +3c =1

X(2) = a(3)^2 + 2b(3)^2 + c(4) (3)^2

3= 9a +12b +36c

X(3) = a(27) + 81b + 243c =5

so after solving the equations we get a=61/135 , b=-2/15 c=2/135