sorting ( / ( + / this equation is valid for any that is power of so we may also write ( / ( / + / / and ( / ( / + / / ( ( + now add up all the equations this means that we add all of the terms on the left-hand side and set the result equal to the sum of all of the terms on the right-hand side observe that the term ( / )/( / appears on both sides and thus cancels in factvirtually all the terms appear on both sides and cancel this is called telescoping sum after everything is addedthe final result is ( (nlog because all of the other terms cancel and there are log equationsand so all the at the end of these equations add up to log multiplying through by gives the final answer (nn log ( log nnotice that if we did not divide through by at the start of the solutionsthe sum would not telescope this is why it was necessary to divide through by an alternative method is to substitute the recurrence relation continually on the righthand side we have ( ( / since we can substitute / into the main equation ( / ( ( ( / ) / ( / we have ( ( / againby substituting / into the main equationwe see that ( / ( ( / / ( / so we have ( ( /