Đáp án: $B=\dfrac2{81}$
Giải thích các bước giải:
Ta có:
$B=\dfrac{1}{10\cdot 9}+\dfrac{1}{18\cdot13}+\dfrac{1}{26\cdot 17}+...+\dfrac{1}{802\cdot 405}$
$\to \dfrac12B=\dfrac{1}{10\cdot 9\cdot2}+\dfrac{1}{18\cdot13\cdot2}+\dfrac{1}{26\cdot 17\cdot2}+...+\dfrac{1}{802\cdot 405\cdot2}$
$\to \dfrac12B=\dfrac{1}{10\cdot 18}+\dfrac{1}{18\cdot26}+\dfrac{1}{26\cdot 34}+...+\dfrac{1}{802\cdot 810}$
$\to 4B=\dfrac{8}{10\cdot 18}+\dfrac{8}{18\cdot26}+\dfrac{8}{26\cdot 34}+...+\dfrac{8}{802\cdot 810}$
$\to 4B=\dfrac{18-10}{10\cdot 18}+\dfrac{26-18}{18\cdot26}+\dfrac{34-26}{26\cdot 34}+...+\dfrac{810-802}{802\cdot 810}$
$\to 4B=\dfrac1{10}-\dfrac1{18}+\dfrac1{18}-\dfrac{1}{26}+\dfrac1{26}-\dfrac1{34}+...+\dfrac1{802}-\dfrac1{810}$
$\to 4B=\dfrac1{10}-\dfrac{1}{810}$
$\to 4B=\dfrac8{81}$
$\to B=\dfrac2{81}$
