Ta có : 

\(A=\frac{101}{1}+\frac{100}{2}+\frac{99}{3}+...+\frac{1}{101}\)

\(A=\left(101-1-...-1\right)+\left(\frac{100}{2}+1\right)+\left(\frac{99}{3}+1\right)+...+\left(\frac{1}{101}+1\right)\)

\(A=\frac{102}{102}+\frac{102}{2}+\frac{102}{3}+...+\frac{102}{101}\)

\(A=102\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{101}+\frac{1}{102}\right)\)

\(\Rightarrow\)\(\frac{A}{B}=\frac{102\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{102}\right)}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{102}}=\frac{102}{1}=102\)

Vậy \(\frac{A}{B}=102\)

Chúc bạn học tốt ~

\(A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{99}}\)

\(\Leftrightarrow3A=1+\frac{1}{3}+\frac{1}{3^{^2}}+...+\frac{1}{3^{98}}\)

\(\Leftrightarrow3A-A=1-\frac{1}{3^{99}}\)

\(\Leftrightarrow2A=1-\frac{1}{3^{99}}\)

\(\Leftrightarrow A=\left(1-\frac{1}{3^{99}}\right)\div2\)

a) = 3/3 x ( -24/54 +45/54 ) : 7/12

   = 1 x 21/54 x 12/7

   = 18/27 

( hiện tại mik chỉ lm đc thế này thui. thông cảm nk )

S = \(\frac12\times\frac13\) + \(\frac13\times\frac14\) + \(\frac14\times\frac15\) + \(\frac15\times\frac16\) + \(\frac17\times\frac18\) + \(\frac18\times\frac19\)

S = \(\frac12\) - \(\frac13\) + \(\frac13\) - \(\frac14\) + \(\frac14\) - \(\frac15\) + \(\frac15\) - \(\frac16\) + \(\frac17\) - \(\frac18\) + \(\frac18\) - \(\frac19\)

S = \(\frac12\) - \(\frac19\)

S = \(\frac{9}{18}-\frac{2}{18}\)

S = \(\frac{7}{18}\)

Đặt A=1/3-2/3^2+3/3^3-4/3^4+...+99/3^99+100/3^100

=>A<1/16

3A=1-2/3+3/3^2-4/3^3+...+99/3^98+100/3^99

=>3A-A=(1-2/3+3/3^2-4/3^3+...+99/3^98+100/3^99)-(1/3-2/3^2+3/3^3-4/3^4+...+99/3^99+100/3^100)

2A=5/3^2-7/3^3+1/3^99-100/3^100

2A=1/3^2(5-7/3+1/3^97-100/3^98)

A=1/18.(8/3+1/3^97-100/3^98)

A=1/54.(8+1/3^96-100/3^97)

Vì 1/54<1/16

=>A<1/16(đpcm)