Question

Tìm n , biết :

a) \(\frac{1}{2}\)+\(\frac{1}{6}\)+\(\frac{1}{12}\)+ ...............  +\(\frac{1}{n.\left(n+1\right)}\)= \(\frac{49}{50}\)

b) \(\frac{2}{3}\)+\(\frac{2}{15}\)+\(\frac{2}{35}\)+ ...............  + \(\frac{2}{\left(2n-1\right).\left(2n+1\right)}\)= \(\frac{50}{51}\)

Answer 1

\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{n\left(n+1\right)}=\frac{49}{50}\)

\(\Rightarrow\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{n\left(n+1\right)}=\frac{49}{50}\)

\(\Rightarrow\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{n}-\frac{1}{n+1}=\frac{49}{50}\)

\(\Rightarrow1-\frac{1}{n+1}=\frac{49}{50}\)

\(\Rightarrow\frac{1}{n+1}=\frac{1}{50}\)

\(\Rightarrow n+1=50\)

\(\Rightarrow n=49\)

\(\frac{2}{3}+\frac{2}{15}+\frac{2}{35}+...+\frac{2}{\left(2n-1\right)\left(2n+1\right)}=\frac{50}{51}\)

\(\Rightarrow\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+...+\frac{2}{\left(2n-1\right)\left(2n+1\right)}=\frac{50}{51}\)

\(\Rightarrow\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2n-1}-\frac{1}{2n+1}=\frac{50}{51}\)

\(\Rightarrow\frac{1}{1}-\frac{1}{2n+1}=\frac{50}{51}\)

\(\Rightarrow\frac{1}{2n+1}=\frac{1}{51}\)

\(\Rightarrow2n+1=51\)

\(\Rightarrow2n=50\)

\(\Rightarrow n=25\)