Salut :)
Daca observi, putem scrie numarul [tex]\frac{1}{2*3}[/tex] ca [tex]\frac{3-2}{2*3}[/tex] sau [tex]\frac{3}{2*3} -\frac{2}{2*3}[/tex] care este [tex]\frac{1}{2}-\frac{1}{3}[/tex] . Acelasi lucru facem la fiecare termen,iar la urma o sa se reduca majoritatea.
[tex](\frac{1}{2*3}+ \frac{1}{3*4}+\frac{1}{4*5}+...+\frac{1}{31*32})-\frac{15}{32}=\\ \\ \\ (\frac{3-2}{2*3}+\frac{4-3}{3*4}+\frac{5-4}{4*5}+...+\frac{32-31}{31*32})-\frac{15}{32}=\\ \\ \\(\frac{3}{2*3}-\frac{2}{2*3}+\frac{4}{3*4}-\frac{3}{3*4}+\frac{5}{4*5}-\frac{4}{4*5}+...+\frac{32}{31*32}-\frac{31}{31*32})-\frac{15}{32}=\\ \\ (\frac{1}{2} -\frac{1}{3} +\frac{1}{3} -\frac{1}{4} +\frac{1}{4} -\frac{1}{5} +...+\frac{1}{31} -\frac{1}{32} )-\frac{15}{32} = \\ \\\frac{16-1}{32} -\frac{15}{32}=0\\[/tex]
