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a) \(\frac{3}{4\times9}+\frac{3}{9\times14}+...+\frac{3}{54\times59}+\frac{3}{59\times64}\)
\(=\frac{3}{5}\times\left(\frac{5}{4\times9}+\frac{5}{9\times14}+...+\frac{5}{59\times64}\right)\)
\(=\frac{3}{5}\times\left(\frac{9-4}{4\times9}+\frac{14-9}{9\times14}+...+\frac{64-59}{59\times64}\right)\)
\(=\frac{3}{5}\times\left(\frac{1}{4}-\frac{1}{9}+\frac{1}{9}-\frac{1}{14}+...+\frac{1}{59}-\frac{1}{64}\right)\)
\(=\frac{3}{5}\times\left(\frac{1}{4}-\frac{1}{64}\right)\)
\(=\frac{9}{64}\)
b) \(\frac{2}{8\times11}+\frac{2}{11\times14}+...+\frac{2}{23\times26}+\frac{2}{26\times29}\)
\(=\frac{2}{3}\times\left(\frac{3}{8\times11}+\frac{3}{11\times14}+...+\frac{3}{26\times29}\right)\)
\(=\frac{2}{3}\times\left(\frac{11-8}{8\times11}+\frac{14-11}{11\times14}+...+\frac{29-26}{26\times29}\right)\)
\(=\frac{2}{3}\times\left(\frac{1}{8}-\frac{1}{11}+\frac{1}{11}-\frac{1}{14}+...+\frac{1}{26}-\frac{1}{29}\right)\)
\(=\frac{2}{3}\times\left(\frac{1}{8}-\frac{1}{29}\right)\)
\(=\frac{7}{116}\)
\(\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+...+\frac{1}{31.35}+\frac{1}{35.39}\) ( . là nhân nhé )
\(=\frac{1}{4}.\left[4.\left(\frac{1}{3.7}+\frac{1}{4.11}+\frac{1}{11.15}+...+\frac{1}{31.35}+\frac{1}{35.39}\right)\right]\)
\(=\frac{1}{4}.\left(\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+...+\frac{4}{31.35}+\frac{4}{35.39}\right)\)
\(=\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{31}-\frac{1}{35}+\frac{1}{35}-\frac{1}{39}\right)\)
\(=\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{39}\right)=\frac{1}{4}.\left(\frac{13-1}{39}\right)=\frac{1}{4}.\frac{12}{39}=\frac{1}{4}.\frac{4}{13}=\frac{1}{13}\)
\(A=\frac{2}{3\times7}+\frac{2}{7\times11}+\frac{2}{11\times15}+...+\frac{2}{99\times103}\)
\(2\times A=\frac{4}{3\times7}+\frac{4}{7\times11}+\frac{4}{11\times15}+...+\frac{4}{99\times103}\)
\(=\frac{7-3}{3\times7}+\frac{11-7}{7\times11}+\frac{15-11}{11\times15}+...+\frac{103-99}{99\times103}\)
\(=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{99}-\frac{1}{103}\)
\(=\frac{1}{3}-\frac{1}{103}=\frac{100}{309}\)
\(\Rightarrow A=\frac{50}{309}\)
\(A=\frac{4}{3X7}+\frac{4}{7X11}+\frac{4}{11X15}+...+\frac{4}{100X104}\)
\(=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{100}-\frac{1}{104}\)
\(=\frac{1}{3}-\frac{1}{104}\)
\(=\frac{101}{312}\)
Chúc bạn học giỏi nha!!!
K cho mik với nhé nguyen huu thuong 2005
\(A=\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+...+\frac{4}{100.104}\)
\(A=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{100}-\frac{1}{104}\)
\(A=\frac{1}{3}-\frac{1}{104}=\frac{104}{312}-\frac{3}{312}=\frac{101}{312}\)
Ta có: \(\dfrac{4}{3\cdot7}+\dfrac{4}{7\cdot11}+\dfrac{4}{11\cdot15}+...+\dfrac{4}{23\cdot27}\)
\(=\dfrac{1}{3}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{15}+...+\dfrac{1}{23}-\dfrac{1}{27}\)
\(=\dfrac{1}{3}-\dfrac{1}{27}=\dfrac{8}{27}\)
\(\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+...+\frac{4}{100.104}\)
\(=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+...+\frac{1}{100}-\frac{1}{104}\)
\(=\frac{1}{3}-\frac{1}{104}=\frac{104}{312}-\frac{3}{312}=\frac{101}{312}\)
\(\frac{3}{2\times5}+\frac{2}{5\times8}+\frac{3}{8\times11}+...+\frac{3}{602\times605}\)
\(=\frac{5-2}{2\times5}+\frac{8-5}{5\times8}+\frac{11-8}{8\times11}+...+\frac{605-602}{602\times605}\)
\(=\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+...+\frac{1}{602}-\frac{1}{605}\)
\(=\frac{1}{2}-\frac{1}{605}=\frac{603}{1210}\)
\(\frac{4}{3\times7}+\frac{5}{7\times12}+\frac{1}{12\times13}+\frac{2}{13\times15}\)
\(=\frac{7-4}{3\times7}+\frac{12-7}{7\times12}+\frac{13-12}{12\times13}+\frac{15-13}{13\times15}\)
\(=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{12}+\frac{1}{12}-\frac{1}{13}+\frac{1}{13}-\frac{1}{15}\)
\(=\frac{1}{3}-\frac{1}{15}=\frac{4}{15}\)