The problem is given below: For two sequences of complex numbers $\{a_0, a_1, \cdots, a_n, \cdots\}$ and $\{b_0, b_1, \cdots, b_n, \cdots\}$ show that the following relations are equivalent: $$a_n = \sum_{k = 0}^n{b_k}\ for\ all\ n \Leftrightarrow b_n = \sum_{k = 0}^n{(-1)^{k + n} a_k}\ for\ all\ n.$$
I was trying to learn Mobius Inversion Formula and Multiplicative functions and I found this problem. I understand how to use the formula to deal with something like $\sum_{d|n} f(d)$, but I didn't get how to solve things like $\sum_{d=0}^n f(d)$. What's the strategy when we face situations like this?
Thanks in advance.