Answer
When $n = 1$, there are 2 quantum states.
When $n = 2$, there are 8 quantum states.
When $n = 3$, there are 18 quantum states.
Work Step by Step
When $n = 1$:
$n = 1, l=0, m = 0, m_s = -\frac{1}{2}$
$n = 1, l=0, m = 0, m_s = \frac{1}{2}$
There are 2 quantum states.
When $n = 2$:
$n = 2, l=0, m = 0, m_s = -\frac{1}{2}$
$n = 2, l=0, m = 0, m_s = \frac{1}{2}$
$n = 2, l=1, m = -1, m_s = -\frac{1}{2}$
$n = 2, l=1, m = -1, m_s = \frac{1}{2}$
$n = 2, l=1, m = 0, m_s = -\frac{1}{2}$
$n = 2, l=1, m = 0, m_s = \frac{1}{2}$
$n = 2, l=1, m = 1, m_s = -\frac{1}{2}$
$n = 2, l=1, m = 1, m_s = \frac{1}{2}$
There are 8 quantum states.
When $n = 3$:
$n = 3, l=0, m = 0, m_s = -\frac{1}{2}$
$n = 3, l=0, m = 0, m_s = \frac{1}{2}$
$n = 3, l=1, m = -1, m_s = -\frac{1}{2}$
$n = 3, l=1, m = -1, m_s = \frac{1}{2}$
$n = 3, l=1, m = 0, m_s = -\frac{1}{2}$
$n = 3, l=1, m = 0, m_s = \frac{1}{2}$
$n = 3, l=1, m = 1, m_s = -\frac{1}{2}$
$n = 3, l=1, m = 1, m_s = \frac{1}{2}$
$n = 3, l=2, m = -2, m_s = -\frac{1}{2}$
$n = 3, l=2, m = -2, m_s = \frac{1}{2}$
$n = 3, l=2, m = -1, m_s = -\frac{1}{2}$
$n = 3, l=2, m = -1, m_s = \frac{1}{2}$
$n = 3, l=2, m = 0, m_s = -\frac{1}{2}$
$n = 3, l=2, m = 0, m_s = \frac{1}{2}$
$n = 3, l=2, m = 1, m_s = -\frac{1}{2}$
$n = 3, l=2, m = 1, m_s = \frac{1}{2}$
$n = 3, l=2, m = 2, m_s = -\frac{1}{2}$
$n = 3, l=2, m = 2, m_s = \frac{1}{2}$
There are 18 quantum states.
