\(\frac{3}{1^2.2^2}+\frac{5}{2^2.3^2}+\frac{7}{3^2.4^2}+.....+\frac{19}{9^2.10^2}\)

\(=\frac{2^2-1^2}{1^2.2^2}+\frac{3^2-2^2}{2^2.3^2}+\frac{4^2-3^2}{3^2.4^2}+......+\frac{10^2-9^2}{9^2.10^2}\)

\(=\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{3^2}-\frac{1}{4^2}+.....+\frac{1}{9^2}-\frac{1}{10^2}\)

\(=\frac{1}{1^2}-\frac{1}{10^2}=1-\frac{1}{10^2}<1\left(đpcm\right)\)

Vì ko hiểu bài lăng nhăng

\(\frac{123}{35}+\frac{34}{35}-\frac{20}{35}\)

= \(\frac{157}{35}-\frac{20}{35}\)

= \(\frac{137}{35}\)