  Working in homotopy type theory, we introduce the notion of $n$-exactness for
a short sequence $F\to E\to B$ of pointed types, and show that any fiber
sequence $F\hookrightarrow E \twoheadrightarrow B$ of arbitrary types induces a
short sequence $\|F\|_{n-1} \to \|E\|_{n-1} \to \|B\|_{n-1}$ that is $n$-exact
at $\|E\|_{n-1}$. We explain how the indexing makes sense when interpreted in
terms of $n$-groups, and we compare our definition to the existing definitions
of an exact sequence of $n$-groups for $n=1,2$. As the main application, we
obtain the long $n$-exact sequence of homotopy $n$-groups of a fiber sequence.
