Xét vế trái : \(T=\frac{1}{5}+\frac{1}{13}+\frac{1}{25}+...+\frac{1}{221}\)

Ta có : \(T< \frac{1}{5}+\frac{1}{12}+\frac{1}{24}+...+\frac{1}{220}\)

               \(=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{6}+\frac{1}{12}+...+\frac{1}{110}\right)=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{10.11}\right)\)

                \(=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{10}-\frac{1}{11}\right)\)

               \(=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{2}-\frac{1}{11}\right)< \frac{1}{5}+\frac{1}{4}\Rightarrow T< \frac{9}{20}\)

Đặt \(A=\frac15+\frac{1}{13}+\frac{1}{25}+\cdots+\frac{1}{10^2+11^2}\)

\(=\frac15+\frac{1}{13}+\frac{1}{25}+\cdots+\frac{1}{100+121}\)

\(=\frac15+\frac{1}{13}+\frac{1}{25}+\cdots+\frac{1}{221}\)

=>\(A<\frac15+\frac{1}{12}+\frac{1}{24}+\cdots+\frac{1}{220}\)

=>\(A<\frac15+\frac12\left(\frac16+\frac{1}{12}+\cdots+\frac{1}{110}\right)\)

=>\(A<\frac15+\frac12\left(\frac12-\frac13+\frac13-\frac14+\cdots+\frac{1}{10}-\frac{1}{11}\right)\)

=>\(A<\frac15+\frac12\left(\frac12-\frac{1}{11}\right)=\frac15+\frac12\cdot\frac{9}{22}=\frac15+\frac{9}{44}\)

=>\(A<\frac{44}{220}+\frac{45}{220}=\frac{89}{220}\)

=>\(A<\frac{99}{220}=\frac{9}{20}\)

đơn giải thôi nhưng mình ko bấn fx đc

​

xét vế trái : \(\frac{1}{5}+\frac{1}{13}+\frac{1}{25}+...+\frac{1}{221}\)

ta có : \(T< \frac{1}{5}+\frac{1}{12}+\frac{1}{24}+...+\frac{1}{220}\)

             \(=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{6}+\frac{1}{12}+...+\frac{1}{110}\right)=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{10.11}\right)\)

             \(=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{10}-\frac{1}{11}\right)\)

              \(=\frac{1}{5}+\frac{1}{2}\left(\frac{1}{2}-\frac{1}{11}\right)< \frac{1}{5}+\frac{1}{4}\Rightarrow T< \frac{9}{20}\)

cíu tui trời ơi

Mình trả lời ở đây rồi bạn tham khảo nhé: Câu hỏi của Nguyễn Hương Giang

Ta thấy:T<1/5+1/12+12/24+...+1/220=1/5+1/2(1/6+1/12+...+1/110)

=1/5+1/2(1/2.3+1/3.4+...+1/11.10)

=1/5+1/2(1/2-1/3+1/3-1/4+....+1/10-1/11)

=1/5+1/2(1/2-1/11)

=1/5+9/44

=89/220

mà 89/220<99/220=9/20

=>T<9/20

vậy T<9/20

đừng bảo bn nào lớp tớ là tớ dùng olm nhé