If you are asking "Can I go from to ?" the answer is "that depends".
It depends upon exactly what that sum is- what values of r is it over?
If you have , then letting n= r+1, so that r= n- 1, f(r)- f(r+ 1) becomes f(n- 1)- f(n). But when r= a, n= a+ 1 and when r= b, n= b+ 1. So we have . Of course we could change the "dummy index" at will so we can write .
But notice that the limits on the sum have changed.
Of course, if the sum were from to that wouldn't matter.