Well, to check your answer, simply try to plug in the result you got in the given difference equation. I think your solution is not 100% correct. The general form of the solution should be . Plug in the given equation and compare to find a,b and c. To find the particular solution, you have to put k = 0 in the general solution. We have to get 1. So, you find the missing parameter. For example, assuming your answer is correct, and we know A(0)=1, then: . Therefore, .
hope this helps