Question

Afla numarul n ∈N* astfel incat:1/1·2+1/2·3+...+1/n(n+1)=100/101

Answer 1

[tex] \frac{1}{1\cdot 2} = \frac{1}{1} - \frac{1}{2} \\ \frac{1}{2\cdot 3} = \frac{1}{2} - \frac{1}{3} \\ \frac{1}{n\cdot (n+1)} = \frac{1}{n} - \frac{1}{n+1} \\ \frac{1}{1\cdot 2}+ \frac{1}{2\cdot 3}+..+ \frac{1}{n\cdot( n+1)} =\\ =\frac{1}{1} - \frac{1}{2}+\frac{1}{2} - \frac{1}{3}+...+\frac{1}{n} - \frac{1}{n+1}=1 - \frac{1}{n+1}\\ 1 - \frac{1}{n+1}= \frac{100}{101} \\ \frac{1}{n+1}=1-\frac{100}{101}\\ \frac{1}{n+1}= \frac{1}{101} \\ n+1=101\\ n=100[/tex]