I don't know it it's a proper question but still as it is troubling me a lot now therefore I am posting this doubt.
Question: how to find sum of
$$Z= \sum (r)^t $$ (from r=1 to r=n(any natural number)) Where t is a constant for any particular series. For e.g. for$$ t=1 => Z =n(n+1)÷2.$$ my approach: I know how it can be done for t=1,2,3 by considering identity $$r^{t+1} -(r-1)^{t+1}=......$$//simplification
But this method is too lengthy to be able to use it for higher value of t. I came across a way while surfing on net that required Bernoulli numbers and some other alien things(for me)...which was far to be proved as helpful.
problem: but for higher value of t it becomes sisyphean task. So I require a simple, elegant, universal and effective method.