I've found this problem in "Discrete Mathematics and Application" by Kenneth Rosen. I also found an explanation of the way of solving this problem which goes like that,
Case 1: Start with a 1 – how many are these? If we remove the 1, then these strings are just the number of such strings of ܽan-1. Why? Because if you remove the 1, then all these strings must make the set ofܽ an - 1, i.e., strings of length ݊ − 1 that do not have two consecutive 0s.
Case 2: Start with a 0 – how many are these? These strings must also have a 1 at the second position. The number of such strings is ܽan-2. Why? Because if remove the last two digits “10”, these strings must make the set of ܽan-2, i.e., strings of length ݊n − 2 that do not have two consecutive 0s.
But i found it too complex, can anyone please make this a little bit easier ?