$\frac{1}{26}$

$\frac{1}{26} = 0.0384615384615384615....$

And so we have: $a_{1} = 0$ $a_{2} = 3$ $a_{3} = 8$ $a_{4} = 4$ $a_{5} = 6$ $a_{6} = 1$ $a_{7} = 5$ and so on.

As you can see, the digits 3, 8, 4, 6, 1, and 5 are repeating every six digits. We'll be using the same principle we use in simplifying $i^n$ to solve this problem. NOTE that the first term, which is 0, does not repeat, therefore instead of dividing 2626 by 7, we will divide 2625 by 6 . Doing so will give us a remainder of 3.

This means that we're looking for the third repeating digit which is 4 . As such, $a_{2626} = 4$