## Total Volume of Concentric Shells With Radii in an Arithmetic Sequence

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{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}

on using using these identities