Question

Tính giá trị biểu thức:

B=\(\dfrac{5}{1\times2}+\dfrac{13}{2\times3}+\dfrac{25}{3\times4}+\dfrac{41}{4\times5}+...+\dfrac{181}{9\times10}\)

Answer 1

\(B=\dfrac{5}{1.2}+\dfrac{13}{2.3}+\dfrac{25}{3.4}+\dfrac{41}{4.5}+...+\dfrac{181}{9.10}\)

\(=\left(\dfrac{1}{1.2}+\dfrac{4}{1.2}\right)+\left(\dfrac{1}{2.3}+\dfrac{12}{2.3}\right)+\left(\dfrac{1}{3.4}+\dfrac{24}{3.4}\right)+...+\left(\dfrac{1}{9.10}+\dfrac{180}{9.10}\right)\)

\(\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{9.10}\right)+\left(\dfrac{4}{1.2}+\dfrac{12}{2.3}+...+\dfrac{180}{9.10}\right)\)

\(=\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{9}-\dfrac{1}{10}\right)+\left(2+2+...+2\right)\)

\(=1-\dfrac{1}{10}+\left(2.9\right)\)

\(=1-\dfrac{1}{10}+18\)

\(=\dfrac{9}{10}+18\)

\(=18\dfrac{9}{10}\)