For every nitely generated group G, assume Gn stands for direct product of n copies of G. Then the sequence fd(Gn)gn1 is called the growth sequence of G, where d(Gn) is the minimum number of generators of Gn. This paper is devoted to consider the growth sequences of G, when G = An1 An2 


Ank is the free products of alternating groups An1 ;An2 ; : : : ;Ank , where ni  5 and 1  i  k. In fact, we prove that d((An1  An2 


 Ank )t) = 2k for all 1  t  h(2;An1 )h(2;An2 )


h(2;Ank ), where h(2;Ani ) is the maximum number m such that Am ni can be generated by two elements. We give several examples and two conjectures at the end.

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