解:$(2)$根据$a_{1} = - 3,$$a_{2}=\frac {1}{4},$$a_{3}=\frac {4}{3},$$a_{4} = - 3,$···,
可知$a_{1},a_{2},a_{3},$···,这列数每三个数为一个循环。
$ $因为$2026\div 3 = 675...1,$$a_{1} + a_{2} + a_{3}=-3+\frac {1}{4}+\frac {4}{3}=-\frac {17}{12},$
$ $所以$-\frac {17}{12}×675 = -\frac {3825}{4}。$
$ $所以$a_{1} + a_{2} + a_{3} + a_{4}+...+a_{2023}+a_{2024}+a_{2025}+a_{2026}=-\frac {3825}{4}+(-3)=-\frac {3837}{4}。$