hi could i get some help with exsplaining what i should have done what would be good examples for proof by cases here thank u

Claim: Let n be any integer. Then n(n+1) is even. Prove this claim using proof by cases. niseven i definedas kz,n=2k where z= set of fallintegers WeAssure that nisemen. Thus we know that n=2k for some integerk n2=4k2=2(2k2) Therefor n2 isevensince its 2 timeso f2h2 Whichisaninteger n(n+1) iseven, n2+n= even ifnisan evenindeger than n2+n isalso even if nisanoddinteover then n2+n isalso even? Thenefor for anyvalue of integerevenorodd the resultinll be ewen. Therefor n(n+1) is even forall integers. I don't su a proof my cases here. please revion some examples