Answer
See answers.
Work Step by Step
Use the results of problem 28-54. For each value of $\ell$, the number of possible states is $2(2\ell + 1)$.
For a given n, $0\leq \ell \leq (n-1)$. Therefore, the number of possible states for a value of n is given by the following sum.
$$\sum_{\ell=0}^{n-1}2(2\ell+1)$$
$$=4\sum_{\ell=0}^{n-1}\ell+\sum_{\ell=0}^{n-1}2$$
Simplify the summation using standard formulas from a pre-calculus textbook.
$$=4\left(\frac{n(n-1)}{2}\right)+2n$$
Simplify further to obtain the desired result.
$$=4\left(\frac{n^2-n}{2}\right)+2n$$
$$=(2n^2-2n)+2n = 2n^2$$
