Question

va roooog ... exercitiul 4 . Mulțumesc Anticipat Dau Coronita, Thanks

Answer 1

Sa analizam formulele:

$\displaystyle\\ \frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$

Formula asta nu putem s-o folosim deoarece diferenta dintre factorii de la numitor este 2.

Fractiile din din paranteza sunt de forma:

$\displaystyle\\ \frac{1}{n(n+2)}$

Ca sa putem aplica formula trebuie sa avem 2 in loc de 1 la numarator.

Rezolvam parenteza:

$\displaystyle\\ \left(\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+\frac{1}{7\cdot9}+...+\frac{1}{61\cdot63}\right)=\\\\ =\frac{1}{2}\left(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+\frac{2}{7\cdot9}+...+\frac{2}{61\cdot63}\right)=\\\\ =\frac{1}{2}\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{61}-\frac{1}{63}\right)=\\\\ =\frac{1}{2}\left(\frac{1}{1}-\frac{1}{63}}\right)=\frac{1}{2}\cdot \frac{62}{63}= \frac{31}{63}$

Acum rezolvam ecuatia:

$\displaystyle\\ x\cdot \frac{31}{63}=2-\frac{64}{63}\\\\ x\cdot \frac{31}{63}=\frac{2\cdot63-64}{63}\\\\ \frac{31x}{63}=\frac{126-64}{63}\\\\ \frac{31x}{63}=\frac{62}{63}~~~\Big| \cdot 63\\\\ 31x = 62\\\\ x=\frac{62}{31}\\\\ \boxed{x = 2} \text{Raspuns corect: }~~\boxed{D}$