Prove that ?n/2? ? n/2? = ?n2/4? for all integers n.
Solution:Step-1: In this problem we need to prove that for all integers n………(1) NOTE: The symbols for floor and ceiling are like the square brackets [ ] with the top or bottom part missing. Example : Case(1) : If n is an even integer. Let us consider , n = 2a then , since from(1)That implies , That implies , Therefore , Is true for all even integers n .Step-2: Case(2): If n is an odd integer. Let us consider , n = 2a+ 1 then , since from(1)That implies , That implies , Therefore , Is true for all odd integers n .If n = 0 , then the given statement is true.Therefore , from case(1) and case(2) is true for all values of n.