Prove that:
\(1< \dfrac{6}{15}+\dfrac{6}{16}+...+\dfrac{6}{19}< 2\)
\(\dfrac{7}{12}< \dfrac{1}{21}+\dfrac{1}{22}+...+\dfrac{1}{40}< \dfrac{5}{6}\)
\(\dfrac{1}{2}< \dfrac{1}{51}+\dfrac{1}{52}+...+\dfrac{1}{100}< 1\)
Find \(x\in Z\):
\(\dfrac{3}{2x+1}+\dfrac{10}{4x+2}-\dfrac{6}{6x+3}=\dfrac{12}{26}\)
Find \(x\in Z\) :
\(\dfrac{-x}{2}+\dfrac{2x}{3}+\dfrac{x+1}{4}+\dfrac{2x+1}{6}=\dfrac{8}{3}\)
Calculate:
\(-\dfrac{1}{3}+\dfrac{1}{3^2}-\dfrac{1}{3^3}+...+\dfrac{1}{3^{100}}-\dfrac{1}{3^{101}}\)
\(\dfrac{1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{999}}{\dfrac{1}{1.999}+\dfrac{1}{3.997}+...+\dfrac{1}{997.3}+\dfrac{1}{999.1}}\)
\(1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2006}}\)
How many fractions have the structure \(\dfrac{a}{11}\) satisfy this operation: \(0\le\dfrac{a}{11}\le99.\)
\(C=\dfrac{8-\dfrac{8}{5}+\dfrac{8}{25}-\dfrac{8}{125}}{9-\dfrac{9}{5}+\dfrac{9}{25}-\dfrac{9}{125}}:\dfrac{161616}{151515}\)
Compact:
\(\dfrac{2^2.3^2.4^2.5^2}{28800}\)
We have:
\(P=\dfrac{4n+1}{2n+3}\) . Find \(n\) so that \(P\) is a natural number.
\(\dfrac{2}{7}-\left(\dfrac{3}{8}+\dfrac{9}{7}\right)\)
\(\dfrac{99}{100}>\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{100^2}>\dfrac{99}{202}\)
\(P=\dfrac{2}{1.3}+\dfrac{2}{3.5}+...+\dfrac{2}{\left(2n+1\right)\left(2n+3\right)}\)
Prove that: \(P< 1,\forall n>0\)
\(\dfrac{7x-21}{14x-42}=\dfrac{2}{4}\)
Find the suitable \(x\in N\) satisfy this fraction as a natural number:
\(\dfrac{2n+8}{n+2}\)
How many numbers \(x\) divided by 11 satisfy this equation:
\(999\le x\le1111\)
\(\dfrac{12n+1}{30n+2}:d\) \(\left(d=1\right)\)
\(\dfrac{9}{10!}+\dfrac{9}{11!}+...+\dfrac{9}{1000!}< \dfrac{1}{9!}\)