Can't you just evaluate it and read the last digit ? Or it is specifically stated you must do it the analytic way ?
By the way it is not a sum ... well not completely
I'd split it into 1^3 + 3^3 + 5^3 +... 39^3
and 2^3 + 4^3 +6^3 +... 38^3 and then consider the difference.
Write down the units digits for cubes of odd numbers and then for even numbers. You'll see a pattern!