[tex]\displaystyle 11+12+...+100=5(1000x-1) \\ \\ 11+12+...+100 \\ \\ 100=11+(n-1) \cdot 1 \Rightarrow 100=11+n-1 \Rightarrow n=100-11+1 \Rightarrow \\ \\ \Rightarrow n=90 \\ \\ S_{90}= \frac{2 \cdot 11+(90-1) \cdot 1}{2} \cdot 90 \\ \\ S_{90}= (22+89 \cdot 1) \cdot 45 \\ \\ S_{90}=(22+89) \cdot 45 \\ \\ S_{90}=111 \cdot 45 \\ \\ S_{90}=4995 \\ \\ 11+12+...+100=5(1000x-1) \\ \\ 4995=5(1000x-1) \\ \\ 4995=5000x-5 \\ \\ 5000x=4995+5 \\ \\ 5000x=5000 \\ \\ x= \frac{5000}{5000} \Rightarrow \boxed{x=1}[/tex]
