You could:
(1) use an STL container to keep the numbers, say list<int>
(2) input the number of elements
(3) input the required number of elements, adding them to the container (..push_back)
(4) make an algorithm to sum the container. they are not overly hard (use ..for_each and a structure or class).
(5) append the new sum to the container (..push_back), and remove the first element from the container (..pop_front).
(6) do steps (4) and (5) 10 times.
No, he could not. The use of all C++ standard features is apparently not the topic of introductory programming courses. Besides, for step 4 there is already std::accumulate, and as already discussed, it is not even needed.
I did miss that bit that the number of outputs is a compile-time constant. @tntxtnt: Cool.