Răspunsuri

2014-11-01T09:07:33+02:00
Aducem la acelasi numitor 1-1/4=(4-1)/4=3/4,1-1/9=(9-1)/9=8/9/,1-1/n patrat=npatrat-1/n la patrat
Pn=3/4x8/9-npatrat-1/npatrat
Pn=24/36-npatrat-1/npatrat
Pn=24npatrat-36(n la patrat-1)=24n patrat-36 n patrat+36=-12npatrat+36
-n la patrat+3 daca :la 12
Si dai valori lui n de la n=2,n=3....
daca e cel mai bun raspuns marcheaza-l.multumesc!
Cel mai inteligent răspuns!
2014-11-01T14:29:44+02:00
8.

 P_{n}= (1- \frac{1}{4})(1- \frac{1}{9})....(1- \frac{1}{ n^{2} })   \\  1- \frac{1}{ n^{2} } =  \frac{ n^{2}-1 }{ n^{2} } =  \frac{(n-1)(n+1)}{ n^{2} }   \\  \\  P_{n} =   \frac{(2-1)(2+1)}{4} \frac{(3-1)(3+1)}{9}...... \frac{(n-1)(n+1)}{ n^{2} }    \\  P_{n} =  \frac{1*3}{4} \frac{2*4}{9}...... \frac{(n-2)n}{ (n-1)^{2} }   \frac{(n-1)(n+1)}{ n^{2} }

Se observa factul ca se simplifica termenii . Vom ajunge la urmatoarea formula :

 P_{n} =  \frac{1}{2} \frac{(n+1)}{n} =  \frac{n+1}{2n}
2 5 2