  In [Phys. Rev. Lett. 114, 250401 (2015)] the sharing of non-locality by
multiple observers was demonstrated through the quantum violation of
Clauser-Horne-Shimony-Halt inequality. In this paper we provide a scheme for
sharing of non-locality and non-trivial preparation contextuality sequentially
through the quantum violation of a family of Bell's inequalities where Alice
and Bob perform $2^{n-1}$ and $n$ numbers of measurements of dichotomic
observables respectively. For this, we consider that Alice always performs
projective measurement and multiple Bobs sequentially perform unsharp
measurement. We show that when Bob's choices of measurement settings are
unbiased, maximum two Bobs can sequentially share the non-locality through the
violation of our inequalities. Further, we show that the local bound of the
aforementioned family of inequalities gets reduced if non-trivial preparation
non-contextuality assumptions are further imposed. Then there is a chance to
share the non-trivial preparation contextuality for more number of Bobs than
that of non-locality. We demonstrate that the non-trivial preparation
contextuality can be sequentially shared by arbitrary numbers of Bob for
unbiased choices of his measurement settings.
