If the first and second of the five numbers in continued proportion are $2$ and $8$ respectively, then the fifth number is:

Let the five numbers in continued proportion be $a_1,a_2,a_3,a_4,$ and $a_5$. Continued proportion implies that these numbers form a Geometric Progression ($\text{GP}$).
Given:

* First term ($a_1$) = $2$
* Second term ($a_2$) = $8$

The common ratio ($r$) of the progression is:

$$r=\frac{a_2}{a_1}=\frac{8}{2}=4$$

The formula for the $n$ -th term of a geometric progression is:

$$a_n=a_1\cdot r^{n-1}$$

For the fifth number ($n=5$):

$$a_5=a_1\cdot r^{5-1}$$

$$a_5=2\cdot 4^4$$

$$a_5=2\cdot 256=512$$

Thus, the fifth number is $512$.