We consider a numeration system which is a common generalization of the positional systems introduced by Cantor and Rényi. Number representations are obtained using a composition of \(\beta _k\) -transformations for a given sequence of real bases \(\varvec{{\mathcal {B}}}=(\beta _k)_{k\ge 1}\) , \(\beta _k>1\) . We focus on arithmetical properties of the set of numbers with finite \(\varvec{{\mathcal {B}}}\) -expansion in case that \(\varvec{{\mathcal {B}}}\) is an alternate base, i.e. \(\varvec{{\mathcal {B}}}\) is a periodic sequence. We provide necessary conditions for the so-called finiteness property. We further show a sufficient condition using rewriting rules on the language of representations. The proof is constructive and provides a method for performing addition of expansions in alternate bases. Finally, we give a family of alternate bases that satisfy this sufficient condition. Our work generalizes the results of Frougny and Solomyak obtained for the case when the base \(\varvec{{\mathcal {B}}}\) is a constant sequence.
