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1) A = \(\frac{-15}{19}.\frac{23}{37}+\frac{14}{37}.\frac{15}{19}=\frac{15}{19}.\frac{-23}{37}+\frac{14}{37}.\frac{15}{19}=\frac{15}{19}.\left(\frac{-23}{37}+\frac{14}{37}\right)=\frac{15}{19}.\frac{-9}{37}=\frac{-135}{703}\)
a)\(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.....\frac{9999}{10000}=\)\(\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.....\frac{99.101}{100.100}=\frac{101}{2.100}=\frac{101}{200}\)
b)\(\frac{8}{9}.\frac{15}{16}.\frac{24}{25}.....\frac{3599}{3600}=\frac{2.4}{3.3}.\frac{3.5}{4.4}.\frac{4.6}{5.6}.....\frac{59.61}{60.60}=\frac{2.61}{60}=\frac{61}{30}\)
A= \(\frac{1^2}{1.2}\). \(\frac{2^2}{2.3}\). \(\frac{3^2}{3.4}\). \(\frac{4^2}{5.6}\).
A= \(\frac{1.1}{1.2}\). \(\frac{2.2}{2.3}\). \(\frac{3.3}{3.4}\). \(\frac{4.4}{4.5}\).
A= \(\frac{1.2.3.4}{1.2.3.4}\). \(\frac{1.2.3.4}{2.3.4.5}\).
A= 1. \(\frac{1}{5}\).
A= \(\frac{1}{5}\).
Vậy A= \(\frac{1}{5}\).
B= \(\frac{3}{4}\). \(\frac{8}{9}\). \(\frac{15}{16}\)..... \(\frac{899}{900}\).
B= \(\frac{1.3}{2.2}\). \(\frac{2.4}{3.3}\). \(\frac{3.5}{4.4}\)..... \(\frac{29.31}{30.30}\).
B= \(\frac{1.2.3.....29}{2.3.4.....30}\). \(\frac{3.4.5.....31}{2.3.4.....30}\).
B= \(\frac{1}{30}\). \(\frac{31}{2}\).
B= \(\frac{31}{60}\).
Vậy B= \(\frac{31}{60}\).
\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}...\frac{9999}{10000}=\frac{3.8.15...9999}{4.9.16...10000}=\frac{1.3.2.4.3.5...99.101}{2.2.3.3.4.4...100.100}=\frac{\left(1.2.3...99\right)\left(3.4.5...101\right)}{\left(2.3.4...100\right)\left(2.3.4...100\right)}\)
\(\frac{1.101}{100.2}=\frac{101}{200}\)
\(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.\frac{2499}{2500}=\frac{3.8.15.2499}{4.9.16.2500}\)\(=\frac{14994}{24000}\)
(Thực hiện rút gọn)
# Học tốt #
\(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.\frac{2499}{2500}=\frac{3.8.15.2499}{4.9.16.2500}=\frac{3.15.2499}{4.9.2.2500}\)
Tự rút gọn tiếp đi
\(A=\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot....\cdot\frac{960}{961}\)
\(=\frac{1.3}{2.2}\cdot\frac{2.4}{3.3}\cdot\frac{3.5}{4.4}\cdot.....\cdot\frac{30.32}{31.31}\)
\(=\frac{\left(1.2.3....30\right)\left(3.4.5....32\right)}{\left(2.3.4....31\right)\left(2.3.4...31\right)}\)
\(=\frac{32}{31.2}=\frac{32}{62}=\frac{16}{31}\)
\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{960}{961}\)
\(\Rightarrow A=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.....\frac{30.32}{31.31}\)
\(\Rightarrow A=\frac{1.2.3....30}{2.3.4....31}.\frac{3.4.5.....32}{2.3.4.....31}\)
\(\Rightarrow A=\frac{1}{31}.\frac{32}{2}=\frac{16}{31}\)