up "heads that ise( ( ) thusif we let (nbe shorthand for ( ( ))then we can write the case for as ( )<= ( / bn to convert this relation into closed formlet us iteratively apply this inequality assuming is large sofor exampleafter two applicationst( <= (( / ) ( / ) bn at this pointwe should see that the general case is in other wordsthe expected running time is at most bn times geometric sum whose base is positive number less than thusby proposition (nis (nproposition the expected running time of randomized quick-select on sequence of size is ( )assuming two elements of can be compared in ( time exercises for source code and help with exercisesplease visit java datastructures net reinforcement - suppose is list of bitsthat isn ' and ' how long will it take to sort with the merge-sort algorithmwhat about quick-sortr- suppose is list of bitsthat isn ' and ' how long will it take to sort stably with the bucket-sort algorithmr- give complete justification of proposition -