\(u_n=\frac{n+1}{n-1}u_{n-1}\)
\(u_{n-1}=\frac{n-1+1}{n-1-1}u_{n-2}=\frac{n}{n-2}u_{n-2}\)
\(u_{n-2}=\frac{n-1}{n-3}u_{n-3}\)
...
\(u_2=\frac{2+1}{2-1}u_1\)
Nhân vế với vế:
\(u_nu_{n-1}u_{n-2}...u_2=\frac{\left(n+1\right)n\left(n-1\right)...3}{\left(n-1\right)\left(n-2\right)\left(n-3\right)...1}u_{n-1}u_{n-2}u_{n-3}...u_1\)
\(\Leftrightarrow u_n=\frac{n\left(n+1\right)}{2}u_1=n\left(n+1\right)\)
\(u_n< 100\Rightarrow n^2+n< 100\)
\(\Leftrightarrow n^2+n-100< 0\Rightarrow n\le9\Rightarrow n=\left\{1;2;...;9\right\}\)
