Question

Clasa a 9-a.
Demonstrati urmatoarele egalitati:
c) (1-$\frac{1}{n ^{2} }$)*(1-$\frac{1}{(n+1)^{2} }$)*...*(1-$\frac{1}{(2n-1)^{2} }$)=1-$\frac{1}{2n-1}$ ; oricare ar fi n ∈ |N*

Answer 1

(n² - 1)/n² ·[(n+1)² - 1]/(n+1)² ·[(n+2)² - 1]/(n+2)² ·...........·[(2n-1)² - 1]/(2n-1)² =(n - 1)(n+1)/n² · n(n+2)/(n+1)² · (n +3)(n +1)/(n+2)² ·.....·4n(n-1)/(2n-1)² =
2(n-1)/(2n-1)
observam: 1-1/n² = (n-1)(n+1) / n²
1 - 1/(n+1)² = n(n+2)/(n+1)²
1 - 1/(n+2)²= (n+1)(n+3)/(n+2)²
...............................................
1 - 1/(n+k)² = (n+k-1)(n+k+1)/(n+k)²
1 - 1/(2n-1)² = 2(n-1)·2/(2n-1)²
se simplifica toti factorii si raman 2(n-1)/(2n-1) = 1- 1/(2n-1)