1. Break this into 2 problems:
(your sum) = a mod 2.
(your sum) = b mod 5.
If you can find a and b, you can find the equivalent:
(your sum) = c mod 10. (use Chinese remainder theorem) and you are done.
Now to find b: (finding a is easier)
Remember that the numbers
1,2,3,4,5,6,7,8,9,10... = 1,2,3,4,0,1,2,3,4,0...mod (5).
So your 'n' in the bottom doesn't vary all that much.
Also, remember Fermat's little theorem => if a isn't a multiple of p, then
a^(p-1)=1(p). In other words, things like:
4^4=1(5), 12^12=1(5), etc.
This should simplify your sum a lot.