a) \(\dfrac{2}{1\times4}+\dfrac{2}{4\times7}+\dfrac{2}{7\times10}+...+\dfrac{2}{97\times100}\)

\(=2.\left(\dfrac{1}{1\times4}+\dfrac{1}{4\times7}+\dfrac{1}{7\times10}+...+\dfrac{1}{97\times100}\right)\)

\(=2.\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{97}-\dfrac{1}{100}\right)\)

\(=2.\left(1-\dfrac{1}{100}\right)\)

\(=2.\dfrac{99}{100}\)

\(=\dfrac{99}{50}\)

_____

b) \(\dfrac{3}{1\times5}+\dfrac{3}{5\times9}+\dfrac{3}{9\times13}+...+\dfrac{3}{97\times101}\)

\(=3.\left(\dfrac{1}{1\times5}+\dfrac{1}{5\times9}+\dfrac{1}{9\times13}+...+\dfrac{1}{97\times101}\right)\)

\(=3.\left(1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+...+\dfrac{1}{97}-\dfrac{1}{101}\right)\)

\(=3.\left(1-\dfrac{1}{101}\right)\)

\(=3.\dfrac{100}{101}\)

\(=\dfrac{300}{101}\)

dấu " . "  là dấu nhân nha bn, nãy viết nhanh quên lớp 

\(-\frac{12}{35}\div\frac{7}{11}-\frac{23}{35}\div\frac{7}{11}-\frac{5}{11}\)

\(=\left(-\frac{12}{35}-\frac{23}{35}\right)\div\frac{7}{11}-\frac{5}{11}\)

\(=-1\div\frac{7}{11}-\frac{5}{11}\)

\(=-\frac{11}{7}-\frac{5}{11}\)

\(=-\frac{156}{77}\)

\(2-\frac{11}{9}\div\frac{5}{14}-\frac{7}{9}\times\frac{14}{5}\)

\(=2-\frac{154}{45}-\frac{98}{45}\)

\(=-\frac{64}{45}-\frac{98}{45}\)

\(=-\frac{18}{5}\)

\(S=\frac{1}{4}\times\left(\frac{4}{5\times9}+\frac{4}{9\times13}+\frac{4}{13\times17}+...+\frac{4}{41\times45}\right)\)

\(S=\frac{1}{4}\times\left(\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+\frac{1}{13}-\frac{1}{17}+...+\frac{1}{41}-\frac{1}{45}\right)\)

\(S=\frac{1}{4}\times\left(\frac{1}{5}-\frac{1}{45}\right)\)

\(S=\frac{1}{4}\times\frac{8}{45}\)

\(S=\frac{1\times2}{1\times45}\)

\(S=\frac{2}{45}\)

Vậy \(S=\frac{2}{45}\)

Tk nha bn !!

\(\frac{8}{135}\) nhé

mình nhầm , kết quả là \(\frac{2}{45}\)

a) 1/1x5 + ... + 1/21x25

= 4 x (1-1/5 + 1/5 - 1/9 + ... + 1/21 - 1/25)

= 1/4 x (1 - 1/25)

= 1/4 x 24/25

= 6/25

\(A=\frac{7}{1\cdot5}+\frac{7}{5\cdot9}+...+\frac{7}{17\cdot21}=\)

\(\frac{4}{7}A=\frac{4}{1\cdot5}+\frac{4}{5\cdot9}+...+\frac{4}{17\cdot21}=\)

\(\frac{4}{7}A=\left(1-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{9}\right)+...+\left(\frac{1}{17}-\frac{1}{21}\right)=\)

\(\frac{4}{7}A=1-\frac{1}{21}=\)

\(\frac{4}{7}A=\frac{20}{21}\)

\(A=\frac{20}{21}\div\frac{4}{7}\)

\(A=\frac{20}{21}\times\frac{7}{4}=\frac{140}{84}=\frac{5}{3}\)

\(\frac{7}{1.5}+\frac{7}{5.9}+\frac{7}{9.13}+...+\frac{7}{17.21}\)

\(=\frac{7}{4}.\left(\frac{4}{1.5}+\frac{4}{5.9}+\frac{4}{9.13}+...+\frac{4}{17.21}\right)\)

\(=\frac{7}{4}.\left(1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+...+\frac{1}{17}-\frac{1}{21}\right)\)

\(=\frac{7}{4}.\left(1-\frac{1}{21}\right)\)

\(=\frac{7}{4}.\frac{20}{21}=\frac{5}{3}\)

 1/5 + 1/45 + 1/117 + 1/221 + 1/357+ .... + 1/159197

Đặt A=1/5 + 1/45 + 1/117 + 1/221 + 1/357+ .... + 1/159197

      A= 1/1x5 + 1/5x9 + 1/9x13 + 1/13x17 + .... + 1/397x401

   Ax4=4/1x5 + 4/5x9 + 4/9x13 + 4/13x17 + .... + 4/397x401

   Ax4=(1-1/5) + (1/5-1/9) + (1/9-1/13)+….+(1/397-1/401)

   Ax4=1-1/401

       A=100/401

Mà 1/4=100/400>100/401=A nên A<1/4.