Sa analizam formulele:
[tex] \displaystyle\\
\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1} [/tex]
Formula asta nu putem s-o folosim deoarece diferenta dintre factorii de la numitor este 2.
Fractiile din din paranteza sunt de forma:
[tex] \displaystyle\\
\frac{1}{n(n+2)} [/tex]
Ca sa putem aplica formula trebuie sa avem 2 in loc de 1 la numarator.
Rezolvam parenteza:
[tex] \displaystyle\\
\left(\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+\frac{1}{7\cdot9}+...+\frac{1}{61\cdot63}\right)=\\\\
=\frac{1}{2}\left(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+\frac{2}{7\cdot9}+...+\frac{2}{61\cdot63}\right)=\\\\
=\frac{1}{2}\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{61}-\frac{1}{63}\right)=\\\\
=\frac{1}{2}\left(\frac{1}{1}-\frac{1}{63}}\right)=\frac{1}{2}\cdot \frac{62}{63}= \frac{31}{63} [/tex]
Acum rezolvam ecuatia:
[tex] \displaystyle\\
x\cdot \frac{31}{63}=2-\frac{64}{63}\\\\
x\cdot \frac{31}{63}=\frac{2\cdot63-64}{63}\\\\
\frac{31x}{63}=\frac{126-64}{63}\\\\
\frac{31x}{63}=\frac{62}{63}~~~\Big| \cdot 63\\\\
31x = 62\\\\
x=\frac{62}{31}\\\\
\boxed{x = 2}
\text{Raspuns corect: }~~\boxed{D} [/tex]
