Răspuns:
Mai intai aplicam un artificiu, inmultirea si impartirea fiecarei fractii cu 2 (adica diferenta dintre factorii de la numitor). Asta ne da spatiu de manevra pentru a transforma fiecare fractie in diferenta de doua fractii. Un fel de operatie inversa aducerii la acelasi numitor. Avem nevoie de diferenta pentru a reduce din termeni.
[tex]\frac{1}{1*3} =\frac{1}{2} *\frac{2}{1*3} =\frac{1}{2} *\frac{3-1}{1*3}=\frac{1}{2} *(\frac{3}{1*3} -\frac{1}{1*3})=\frac{1}{2} *(\frac{1}{1} -\frac{1}{3})[/tex]
[tex]\frac{1}{3*5} =\frac{1}{2} *\frac{2}{3*5} =\frac{1}{2} *\frac{5-3}{3*5}=\frac{1}{2} *(\frac{5}{3*5} -\frac{3}{3*5})=\frac{1}{2} *(\frac{1}{3} -\frac{1}{5})[/tex]
[tex]\frac{1}{5*7} =\frac{1}{2} *\frac{2}{5*7} =\frac{1}{2} *\frac{7-5}{5*7}=\frac{1}{2} *(\frac{7}{5*7} -\frac{5}{5*7})=\frac{1}{2} *(\frac{1}{5} -\frac{1}{7})[/tex]
etc.
Suma din exercitiu devine:
[tex]=\frac{1}{2} *(\frac{1}{1} -\frac{1}{3})+\frac{1}{2} *(\frac{1}{3} -\frac{1}{5})+\frac{1}{2} *(\frac{1}{5} -\frac{1}{7})+ ... + \frac{1}{2} *(\frac{1}{2017} -\frac{1}{2019}) =[/tex]
[tex]=\frac{1}{2} *(\frac{1}{1} -\frac{1}{3}+\frac{1}{3} -\frac{1}{5}+\frac{1}{5} -\frac{1}{7}+ ... + \frac{1}{2017} -\frac{1}{2019}) =[/tex]
observam ca termenii se reduc, ramanand doar primul si ultimul
[tex]=\frac{1}{2} *(\frac{1}{1} -\frac{1}{2019}) = \frac{1}{2} *\frac{2019-1}{2019}=\frac{1}{2} *\frac{2018}{2019} =[/tex]
[tex]=\frac{1009}{2019}[/tex]
Explicație pas cu pas:
