8. 阅读下面材料：
将边长分别为$a,a+\sqrt{b},a+2\sqrt{b},a+3\sqrt{b}$的正方形面积分别记为$S_1,S_2,S_3,S_4$.
则$S_2 - S_1 = (a+\sqrt{b})^2 - a^2 = [(a+\sqrt{b}) + a] · [(a+\sqrt{b}) - a]$
$\quad\quad\quad\quad\quad= (2a + \sqrt{b}) · \sqrt{b} = b + 2a\sqrt{b}.$
例如:当$a=1,b=3$时,$S_2 - S_1 = 3 + 2\sqrt{3}.$
根据以上材料解答下列问题:
(1)当$a=1,b=3$时,$S_3 - S_2 =$ $,S_4 - S_3 =$ .
(2)当$a=1,b=3$时,把边长为$a + n\sqrt{b}$的正方形面积记作$S_{n+1}$,其中$n$是正整数,从(1)中的计算结果,你能猜出$S_{n+1} - S_n$等于多少吗? 请证明你的猜想.
(3)当$a=1,b=3$时,令$t_1 = S_2 - S_1,t_2 = S_3 - S_2,t_3 = S_4 - S_3,···,t_n = S_{n+1} - S_n$,且$T = t_1 + t_2 + t_3 + ··· + t_{50}$,求$T$的值.

8. 解:
(1)$9 + 2\sqrt{3}\quad 15 + 2\sqrt{3}$
(2)$S_{n+1} - S_n =6n -3 +2\sqrt{3}$,证明略
(3)当$a = 1,b = 3$时,$T = t_1 + t_2 + t_3 + ··· + t_{50}$
$= S_2 - S_1 + S_3 - S_2 + S_4 - S_3 + ··· + S_{51} - S_{50}$
$= S_{51} - S_1 = (1 + 50\sqrt{3})^2 - 1$
$= 7500 + 100\sqrt{3}.$