Rút gọn biểu thức S, ta có:
\(S=\frac{13}{30}+\frac{13}{15}+\frac{13}{35}+\frac{13}{63}+\frac{13}{99}\)
\(\Leftrightarrow S=\frac{13}{30}+\left(\frac{13}{3\cdot5}+\frac{13}{5\cdot7}+\frac{13}{7\cdot9}+\frac{13}{9\cdot11}\right)\)
Đặt \(P=\frac{13}{3\cdot5}+\frac{13}{5\cdot7}+\frac{13}{7\cdot9}+\frac{13}{9\cdot11}\)
\(\Rightarrow P\cdot\frac{2}{13}=\frac{2}{3.5}+\frac{2}{5\cdot7}+\frac{2}{7\cdot9}+\frac{2}{9\cdot11}\)
\(\Rightarrow P\cdot\frac{2}{13}=\frac{2}{3}-\frac{2}{5}+\frac{2}{5}-\frac{2}{7}+\frac{2}{7}-\frac{2}{9}+\frac{2}{9}-\frac{2}{11}\)
\(\Rightarrow P\cdot\frac{2}{13}=\frac{2}{3}-\frac{2}{11}\)
\(\Rightarrow P\cdot\frac{2}{13}=\frac{16}{33}\)
\(\Rightarrow P=\frac{104}{33}=3\frac{5}{33}\)
Ta có: \(P+1>P+\frac{13}{30}\)
Mà  \(P+\frac{13}{30}=S\)
Còn \(P+1=3\frac{5}{33}+1=4\frac{5}{33}<5\)
\(\Rightarrow S<4\frac{5}{33}<5\)

Vậy đề bài sai.

Bài này sai đề rồi.

Sai rõ 

\(=\frac{13\times10101}{15\times10101}+\frac{13\times10101}{35\times10101}+\frac{13\times10101}{63\times10101}+\frac{13\times10101}{99\times10101}\)

\(=\frac{13}{15}+\frac{13}{35}+\frac{13}{63}+\frac{13}{99}=\frac{13}{2}\times\left(\frac{2}{3\times5}+\frac{2}{5\times7}+\frac{3}{7\times9}+\frac{2}{9\times11}\right)\)

\(=\frac{13}{2}\times\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}\right)=\frac{13}{2}\times\left(\frac{1}{3}-\frac{1}{11}\right)=\frac{13}{2}\times\frac{8}{33}=\frac{52}{33}\)

\(\frac{13.10101}{15.10101}\)+\(\frac{13.10101}{15.10101}\)+\(\frac{13.10101}{63.10101}\)+ \(\frac{13.10101}{99.10101}\)= \(\frac{13}{15}\) +   \(\frac{13}{15}\) + \(\frac{13}{63}\)+  \(\frac{13}{99}\) =\(2\frac{82}{1155}\)

Kết quả của phép tính trên là \(\frac{52}{33}.\)

\(\frac{131313}{151515}+\frac{131313}{353535}+\frac{131313}{636363}+\frac{131313}{999999}\)

=\(\frac{13}{15}+\frac{13}{35}+\frac{13}{63}+\frac{13}{99}\)

=\(13\left(\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}\right)\)

=\(13.\frac{1}{2}\left(\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}\right)\)

=\(\frac{13}{2}.\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}\right)\)

=\(\frac{13}{2}.\left(\frac{1}{3}-\frac{1}{11}\right)\)

= \(\frac{13}{2}.\frac{8}{33}\)

=\(\frac{52}{33}\)

52/33

\(\frac{3}{2}.x-70\frac{10}{11}:\left(\frac{131313}{151515}+\frac{131313}{353535}+\frac{131313}{636363}+\frac{131313}{999999}=-5\right)\)

\(\frac{3}{2}.x-70\frac{10}{11}:\left(\frac{13}{15}+\frac{13}{35}+\frac{13}{63}+\frac{13}{99}\right)=-5\)

\(\frac{3}{2}.x-70\frac{10}{11}:\left[13\times\left(\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}\right)\right]=-5\)

\(\frac{3}{2}.x-70\frac{10}{11}:\left[13\times\left(\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\frac{1}{9.11}\right)\right]=-5\)

\(\frac{3}{2}.x-70\frac{10}{11}:\left[13\times\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}\right)\right]=-5\)

\(\frac{3}{2}.x-70\frac{10}{11}:\left[13\times\left(\frac{1}{3}-\frac{1}{11}\right)\right]=-5\)

\(\frac{3}{2}.x-\frac{45}{2}=-5\)

\(\frac{3}{2}.x=\frac{35}{2}\)

\(x=\frac{35}{3}\)

bài phát sai òi hihi cần mk chỉ chỗ sai ko

x= 80/3 đúng ko các bạn

= 13/15 + 13/35 + 13/63 + 13/99
= 13/ 3×5 + 13/5×7 + 13/7×9 + 13/9×11
= 13 x 1/2( 1/3 – 1/5 + 1/5 – 1/7 +1/7 – 1/9 +1/9 – 1/11)
= 13/2 x ( 1/3 – 1/11)
= 13/2 x 8/33 = 104/66=52/33

\(D=\frac{13}{15}+\frac{13}{35}+\frac{13}{63}+\frac{13}{99}\)

\(D=\frac{13}{3}\times5+\frac{13}{5}\times7+\frac{13}{7}\times9+\frac{13}{9}\times11\)

\(D=13\times\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}\right)\)

\(D=\frac{13}{2}\times\left(\frac{1}{3}-\frac{1}{11}\right)\)

\(D=\frac{13}{2}\times\frac{8}{33}=\frac{104}{66}=\frac{52}{33}\)

\(D=\frac{52}{33}\)

(131313/151515+ 131313/353535+131313/636363+131313/999999)*33

=(13/15+13/35+13/63+13/99)*33

=52/33*33

=52