For a finitely generated group G, we denote Gn as the direct product of n copies of G. The growth sequence of G is the sequence {d(Gn)}n≥1, where d(Gn) is the minimum number of generators of Gn. In this paper, we investigate the growth sequence of G, when G is the free product of alternating groups. In fact, we prove that d  (An ∗ Am) h(2,An)h(k,Am)  ≤ k +2, for all n, m ≥ 5 and k ≥ 2, where h(2,An) is the maximum number t such that d(At n) = 2 and similarly, h(k,Am) is the maximum number s such that d(As m)= k. Moreover, we will consider the case k = 2 and prove that d((An ∗ Am)t ) = 4, for all 1 ≤ t ≤ h(2,An)h(2,Am) and n,m ≥ 5. We have also confirmed the above results by several examples in find section.

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