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Let A and B be finite sets in a commutative group. We bound |A+hB| in terms of |A|, |A+B| and h. We provide a submultiplicative upper bound that improves on the existing bound of Imre Ruzsa by inserting a factor that decreases with h.

We offer a complete answer to the following question on the growth of sumsets in commutative groups. Let h be a positive integer and $A,B₁,...,B_h$ be finite sets in a commutative group. We bound $|A+B₁+...+B_h|$ from above in terms of |A|, |A + B₁|, ..., $|A+B_h|$ and h. Extremal examples, which demonstrate that the bound is asymptotically sharp in all parameters, are furthermore provided.