A sequence is defined by letting and for all integers . Prove that for all integers , .
Base case: S(2)
Let . Then .
Let . Then
Base is true.
Induction step: Assume S(k) and are true. Then
Since and , by definition .
Solving this should lead to that
Therefore is true.
I think I've got the right idea but I'm not completely sure that I've done this correctly. Any help or suggestions are much appreciated!