The countable direct sum
$$
\bigoplus_{n\geq0}\mathbb Z
$$
is abelian and therefore nilpotent of class one. It is not finitely generated, whereas every <polycyclic group> is finitely generated. Hence it is nilpotent but not polycyclic.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 2 / a / 2 / Solution

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A group is polycyclic when it has a subnormal series with cyclic factors. Every subgroup and quotient of a polycyclic group is finitely generated, and extensions of polycyclic groups are polycyclic.


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