  In this article, we consider the $4+n$ dimensional spacetimes among which one
is the four dimensional physical Universe and the other is an n-dimensional
sphere with constant radius in the framework of Lanczos-Lovelock gravity. We
find that the curvature of extra dimensional sphere contributes a huge but
negative energy density provided that its radius is sufficiently small, such as
the scale of Planck length. Therefore, the huge positive vacuum energy, i.e.
the large positive cosmological constant is exactly cancelled out by the
curvature of extra sphere. In the mean time the higher order of
Lanczos-Lovelock term contributes an observations-allowed small cosmological
constant if the number of extra dimensions is sufficiently large, such as
$n\approx{69}$.
