# How can I make this recurrent equation non-recurrent?

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?

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The given equation easily defines $b_k$ in terms of a summation over terms $a_i$ from $i=k$ to infinity. Are you trying to "solve" this relationship to obtain $a_k$ in terms of $b_i$'s? – hardmath Jun 8 '14 at 12:32
@hardmath Yes, that's what I'm trying to do. I just saw myself how trivial this question is. – FUZxxl Jun 8 '14 at 12:34
@FUZxxl: you'd better rephrase your question: there is no $a_k$ on the RHS ! (Just a few $a_i$'s, but no $a_k$ of any kind). – Yves Daoust Jun 8 '14 at 12:36
Obviously, the $b_k$ are the "integral" of the $a_k$, so that the $a_k$ are the "derivative" of the $b_k$. – Yves Daoust Jun 8 '14 at 12:40

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