How do I calculate the HP if it say: 1^11 + 2^11 + 3^11 + 4^11 + ......n^11
or if it is say: 1^21 + 2^21 + 3^21 + ......... + n^21
The question is the raised to index can be any natural No.

You are going to get to some pretty big numbers pretty fast. Further, you are going to have a fairly slow program if you do it by brute force. I'd suggest finding a math textbook and see if there is some easier way to compute these values. I didn't get that far in math, so I have no idea, but there is probably something out there.

research would be your best bet at this point.

There are only two hard things in computer science: cache invalidation, naming things, and off-by-one errors