Answer
$r^{2}\lt x^{2} + y^{2} + z^{2} \lt R^{2}$
Work Step by Step
A sphere with radius $r$ and center the origin has the equation:
$x^{2} + y^{2} + z^{2} = r^{2}$
A sphere with radius $R$ and center the origin has the equation:
$x^{2} + y^{2} + z^{2} = R^{2}$
Since $r \lt R$, all the points between and not on these spheres are the points that are both inside the bigger sphere with radius $R$ and outside the smaller sphere with radius $r$.
The points inside the bigger sphere are represented by:
$x^{2} + y^{2} + z^{2} \lt R^{2}$
The points outside the smaller sphere are represented by:
$x^{2} + y^{2} + z^{2} \gt r^{2}$
Combining these we get:
$r^{2}\lt x^{2} + y^{2} + z^{2} \lt R^{2}$
