Giải thích các bước giải:
$\dfrac{3}{9.14}+\dfrac{3}{14.19}+\dfrac{3}{19.24}+...+\dfrac{3}{(5n-1)(5n+4)}$
`=\frac{3}{5}.(\frac{1}{9}-\frac{1}{14}+\frac{1}{14}-\frac{1}{19}+....+\frac{1}{5n-1}-\frac{1}{5n+4})`
`=\frac{3}{5}.(\frac{1}{9}-\frac{1}{5n+4})`
$=\dfrac{3}{5}.\dfrac{5(n-1)}{9(5n+4)}$
$=\dfrac{n-1}{3(5n+4)}$
$=\dfrac{n-1}{15n+12}$
`=(n+\frac{4}{5}-\frac{9}{5}):(15n+12)`
`=(n+\frac{4}{5}):(15n+12)-\frac{9}{5}:(15n+12)`
$=\dfrac{5n+4}{5}.\dfrac{1}{3(5n+4)}-\dfrac{9}{5}.\dfrac{1}{15n+12}$
$=\dfrac{1}{15}-\dfrac{9}{75n+60}$
$\text{Vì $\dfrac{9}{75n+60} > 0$}$
$\text{nên $\dfrac{1}{15}-\dfrac{9}{75n+60} < \dfrac{1}{15}$}$
$\text{⇒ ĐPCM}$
Chúc bạn học tốt !!!