Suppose we have an arranagement of spherical concentric shells, each of thickness 1, separated by spaces of thickness 1.
The first shell has outer thickness 1 and inner thickness 0. The second shell has outer thickness 3 and inner thickness 2. The third shell has outer thickness 5 and inner thickness 4. Continue in this way, then the k<supth shell has outer thickness
\[2k-1\]
and inner thickness \[2k-2\]
The kth shell will have volume
\[\frac{4 \pi}{3} ((2k-1)^3 - (2k-2)^3) = \frac{4 \pi}{3} ( 12k^2 -18k+7) \]
Adding up the volume of these
\[n\]
shells gives\[\begin{equation} \begin{aligned} V &= \sum^n_1 \frac{4 \pi}{3} ( 12k^2 -18k+7) \\ &= 16 \pi \sum^n_1 k^2 - 24 \pi \sum^n_1 k + \frac{28 \pi}{3} \sum^n_1 1 \\ &= 16 \pi \frac{n}{6}(n+1)(2n+1) - 18 \pi \frac{n(n+1)}{2}+ \frac{28 \pi}{3} n \end{aligned} \end{equation}\]
on using using these identities