Using induction to prove a general form from a recurrence relation

Question

I have the recurrence relation: $a_{n+2} = \dfrac{-a_{n}}{(n+2)(n+1)}$.

I have done some work to identify that two cases emerge: one is n is positive, and one if n is negative.

If n = 2m (even) $a_{n} = a_{2m} = \dfrac{(-1)^m a_0}{(2m)!}$.

If n = 2m+1 (odd) $a_{n} = a_{2m + 1} = \dfrac{(-1)^m a_1}{(2m+1)!}$.

My problem is using induction to show that this is true for ALL n, when I have two cases. How can I show the case for k and k+1? There is no way to know if k and k+1 are even or odd.....Can anyone help?

Thanks!

Answer 1

Make your induction step based on assuming the expression is valid for 2k (use this to prove 2k+2) and 2k+1 (use to prove 2k+3).