Question

(\(\frac{11}{12}\)+\(\frac{11}{12.23}\)+\(\frac{11}{23.34}\)+....+\(\frac{11}{89.100}\)) + x = \(\frac{5}{3}\)

Answer 1

\(\left(\frac{11}{12}+\frac{11}{12.23}+\frac{11}{23.34}+...+\frac{11}{89.100}\right)+x=\frac{5}{3}\)

\(\Leftrightarrow\)\(\left(\frac{1}{1}-\frac{1}{12}\right)+\left(\frac{1}{12}-\frac{1}{23}\right)+\left(\frac{1}{23}-\frac{1}{34}\right)+...+\left(\frac{1}{89}-\frac{1}{100}\right)+x=\frac{5}{3}\)

\(\Leftrightarrow\)\(1-\frac{1}{100}+x=\frac{5}{3}\)

\(\Leftrightarrow\)\(\frac{99}{100}+x=\frac{5}{3}\)

\(\Leftrightarrow\)\(x=\frac{5}{3}-\frac{99}{100}\)

\(\Leftrightarrow\)\(x=\frac{203}{300}\)

Vậy \(x=\frac{203}{300}\)

Answer 2

\(\left(\frac{11}{12}+\frac{11}{12.23}+\frac{11}{23.34}+...+\frac{11}{89.100}\right)+x=\frac{5}{3}\)

\(\Leftrightarrow\)\(\left(1-\frac{1}{12}+\frac{1}{12}-\frac{1}{23}+\frac{1}{23}-\frac{1}{34}+...+\frac{1}{89}-\frac{1}{100}\right)+x=\frac{5}{3}\)

\(\Leftrightarrow\)\(\left(1-\frac{1}{100}\right)+x=\frac{5}{3}\)

\(\Leftrightarrow\)\(\frac{99}{100}+x=\frac{5}{3}\)

\(\Leftrightarrow\)\(x=\frac{5}{3}-\frac{99}{100}=\frac{203}{300}\)

Answer 3

1/11.(11/12+11/12-11/23+...+1/89-1/100)+x=5/3

1/11.(11/12+11/12-1/100)+x=5/3

1/11.547/300+x=5/3

547/3300+x=5/3

x=5/3-547/3300

x=1651/1100