[tex] 4^{*}\\ \\ a) Aici~ai~doar~de~amplificat:\\ \frac{1}{k} -\frac{1}{k+1} =\frac{k+1-k}{k(k+1)} =\frac{1}{k(k+1)},~\forall~k\in~N^{*}.\\ \\ b)~Aici~te~folosesti~de~formula~de~la~a).\\ S=\frac{1}{1}-\frac{1}{2} +\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}=1-\frac{1}{100}=\frac{99}{100} \\ \\ 5^{*}.\\ \\ Fie~t\in~Z~astfel~incat~\frac{x\sqrt{3}+y}{2\sqrt{3}-2}=t\Leftrightarrow~x\sqrt{3}+y=2\sqrt{3}t-2t\\ \\ \sqrt{3}(x-2t)=-y-2t \\ \\ \Rightarrow~\sqrt{3}=\frac{2t+y}{2t-x} \in~Q~(Fals,~\sqrt{3}\notin~Q)\\ \\ \Rightarrow~\left \{ {{2t+y=0} \atop {2t-x=0}} \right. \Leftrightarrow~\left \{ {{y=-2t} \atop {x=2t}} \right. \Rightarrow~x=-y.\\ \\ Deci,~x+y=0. [/tex]
