TR-MO-R1 2017 Problem 18

$n=1,2,3,\ldots$ doğal sayıları için $a_n = 2 - \dfrac{1}{n^2 - \sqrt{n^4 + \frac{1}{4}}}$ ise

$$\frac{1}{\sqrt{a_1}} + \frac{2}{\sqrt{a_2}} + \frac{3}{\sqrt{a_3}} + \cdots + \frac{19}{\sqrt{a_{19}}} + \frac{20}{\sqrt{a_{20}}} = ?$$

A) $\dfrac{\sqrt{761}+1}{4}$ B) $\dfrac{\sqrt{761}-1}{4}$ C) $20$ D) $19$ E) $7$