chap analysis techniques give and prove the closed-form solution for the recurrence relation (nt( ( give and prove the closed-form solution for the recurrence relation (nt( ct( prove by induction that the closed-form solution for the recurrence relation ( ( / nt( is in ohm( log for the following recurrencegive closed-form solution you should not give an exact solutionbut only an asymptotic solution ( using th notationyou may assume that is power of prove that your answer is correct (nt( / for ( using the technique of expanding the recurrencefind the exact closed-form solution for the recurrence relation ( ( / nt( you may assume that is power of section provides an asymptotic analysis for the worst-case cost of function buildheap give an exact worst-case analysis for buildheap for each of the following recurrencesfind and then prove (using inductionan exact closed-form solution when convenientyou may assume that is power of (at(nt( / for ( (bt( ( / for ( use theorem to prove that binary search requires th(log ntime recall that when hash table gets to be more than about one half fullits performance quickly degrades one solution to this problem is to reinsert all elements of the hash table into new hash table that is twice as large assuming that the (expectedaverage case cost to insert into hash table is th( )prove that the average cost to insert is still th( when this reinsertion policy is used given - tree with nodesprove that inserting additional nodes requires ( node splits