chap analysis techniques ( ( / ( / ) ( ( ( / ( / ) ( / )   ( - - this last expression can best be represented by summation as follows - / = - / = from equation we have / - ( / this is the exact solution to the recurrence for power of two at this pointwe should use simple induction proof to verify that our solution is indeed correct example our next example comes from the algorithm to build heap recall from section that to build heapwe first heapify the two subheapsthen push down the root to its proper position the cost isf ( < ( / log let us find closed form solution for this recurrence we can expand the recurrence few times to see that ( < ( / log < [ ( / log / log < [ ( ( / log / log / log