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An $n$-exact category is a pair consisting of an additive category and a class of sequences with $n+2$ terms satisfying certain axioms. We introduce $n$-weakly idempotent complete categories. Then we prove that an additive $n$-weakly idempotent complete category together with the class ${𝒞}_n$ of all contractible sequences with $n+2$ terms is an $n$-exact category. Some properties of the class ${𝒞}_n$ are also discussed.