So sánh A và B

A=\(\frac{10^{11}-1}{10^{12}-1}\)           và         B=\(\frac{10^{10}+1}{10^{11}+1}\)

gợi ý:

nếu \(\frac{a}{b}\)>1 thì \(\frac{a}{b}\)>\(\frac{a+m}{b+m}\)         (m thuộc z)

        \(\frac{a}{b}\)<1 thì \(\frac{a}{b}\)<\(\frac{a+m}{b+m}\)

 A<1\(\) s\(\)uy ra A=\(\frac{10^{11}-1}{10^{12}-1}\)<\(\frac{10^{11}-1+11}{10^{12}-1+11}\)=\(\frac{10^{11}+10}{10^{12}+10}\)

`1.

a/Chứng minh rằng: \(\frac{1}{41}\)+\(\frac{1}{42}\)+\(\frac{1}{43}\)+...+\(\frac{1}{80}\)>\(\frac{7}{12}\)

b/Chứng minh rằng:: \(\frac{1}{41}\)+\(\frac{1}{42}\)+\(\frac{1}{43}\)+...+\(\frac{1}{60}\)>\(\frac{1}{3}\)

c//Chứng minh rằng:: \(\frac{1}{61}\)+\(\frac{1}{62}\)+\(\frac{1}{62}\)+...+\(\frac{1}{80}\)>\(\frac{1}{4}\)

2.Cho phân số\(\frac{a}{b}\)=\(\frac{1}{6}\)+\(\frac{1}{7}\)+\(\frac{1}{8}\)+\(\frac{1}{9}\)+\(\frac{1}{10}\)+\(\frac{1}{11}\)với a,b thuộc N. Chứng minh rằng a chia hết cho 17

\(\frac{1}{6}\)

TÍNH:

a) \(\frac{1}{12}\)+\(\frac{1}{20}\)\(\frac{1}{30}\)+\(\frac{1}{42}\)+\(\frac{1}{56}\)+\(\frac{1}{72}\)+\(\frac{1}{90}\)+\(\frac{1}{110}\)+\(\frac{1}{132}\)

b) (1-\(\frac{1}{7}\)).(1-\(\frac{1}{8}\)).(1-\(\frac{1}{9}\)) .... ( 1-\(\frac{1}{2011}\))

c) ( \(\frac{1}{10}\)-1).(\(\frac{1}{11}\)-1).(\(\frac{1}{12}\)-1) .... ( \(\frac{1}{2012}\)-1)

d) \(\frac{8^2}{7.9}\). \(\frac{9^2}{8.10}\).\(\frac{10^2}{9.11}\).\(\frac{11^2}{10.12}\).\(\frac{12^2}{11.13}\).\(\frac{13^2}{12.14}\).\(\frac{14^2}{13.15}\)

a) \(Cho\)\(S=\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+.....+\frac{1}{60}\)

\(Chứng\) \(minh\) \(\frac{3}{5}< S< \frac{4}{5}\)

b) \(Chứng\)\(minh\)\(\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+.....+\frac{1}{100}>\frac{7}{10}\)

c) \(Chứng\)\(minh\)\(\frac{3}{10}+\frac{3}{11}+\frac{3}{12}+\frac{3}{13}+\frac{3}{14}\) không là số tự nhiên

d) \(Chứng\)\(minh\)\(\frac{1}{5}< D< \frac{1}{10}\) \(với\)\(D=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{99}{100}\)