If there are no things in the array, then the sum is 0. If there is at least one thing, then the sum of the first n of them is the same as the sum of the first n-1 of them plus the n-th one.

  public static int sum(int[] A, int n)
  {
    if(n == 0) return 0;
    else return sum(A, n-1) + A[n-1];
  }