Giải thích các bước giải:
$\dfrac{1}{17}+\dfrac{7}{17.27}+\dfrac{7}{27.37}+.....+\dfrac{7}{1997.2007}$$=\dfrac{1}{17}+\dfrac{7}{17.27}+\dfrac{7}{27.37}+.....+\dfrac{7}{1997.2007}$
$=\dfrac{1}{17}+\dfrac{7}{10}.\dfrac{10}{17.27}+\dfrac{7}{10}.\dfrac{10}{27.37}+.....+\dfrac{7}{10}.\dfrac{10}{1997.2007}$
$=\dfrac{1}{17}+\dfrac{7}{10}.(\dfrac{1}{17}-\dfrac{1}{27})+\dfrac{7}{10}.(\dfrac{1}{27}-\dfrac{1}{37})+.....+\dfrac{7}{10}.(\dfrac{1}{1997}-\dfrac{1}{2007})$
$=\dfrac{1}{17}+\dfrac{7}{10}.(\dfrac{1}{17}-\dfrac{1}{27}+\dfrac{1}{27}-\dfrac{1}{37}+.....+\dfrac{1}{1997}-\dfrac{1}{2007})$
$=\dfrac{1}{17}+ \dfrac{7}{10}.(\dfrac{1}{17}-\dfrac{1}{2007})$
$=\dfrac{1}{17}+ \dfrac{7}{10}.\dfrac{1}{17}-\dfrac{7}{10.2007}$
$=\dfrac{1}{17}.(1+ \dfrac{7}{10})-\dfrac{7}{20070}$
$=\dfrac{1}{17}.\dfrac{17}{10}-\dfrac{7}{20070}$
$=\dfrac{1}{10}-\dfrac{7}{20070} $
$=\dfrac{200}{2007}$
