How can I make this recurrent equation non-recurrent?

Question

I have a recurrent equation that defines a sequence $a_k$ from a sequence $b_k$:

$$a_k = b_k - \sum^\infty_{i=k+1}a_i$$

How can I write this equation without mentioning $a_k$ on the right side?

Answer 1

$$b_k = a_k + \sum_{i=k+1}a_i = \sum_{i=k}a_i$$

$$b_{k-1} - b_k = \sum_{i=k-1}a_i - \sum_{i=k}a_i = a_{k-1}$$