@vv, thanks for the answer, but I am looking for more generic result. Function inside sum can be much more complex.

Is it possible to rewrite a recurrence with a sum to a recurrence wihout a sum automatically with maple?

@Carl Love, thanks for detailed answer!

I think that I formulated the question not entirely correctly. In fact I have a finite series where n is always some power of 2, and the last term will be n/n=1. The sum of such series is 2*n - 1.

In any case, your answer gave me a lot to think about.

@Kitonum, thanks for a quick answer, but I am looking for the answer (2*n - 1) from Maple, i.e. the program should print it directly, based on some input.

I have tried to use rsolversolve({f(0) = n, f(x) = f(x-1)+n/(2^x)}, f); and it outputs the result very similar to the desired. Maybe I should add some constraints or something to get the desired result: 2*n - 1 ?