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a/ Ta co: \(B=3+3^3+3^5+...+3^{1987}+3^{1989}+3^{1991}\)
\(\Rightarrow B=\left(3+3^3+3^5\right)+...+\left(3^{1987}+3^{1989}+3^{1991}\right)\)
\(\Rightarrow B=3\cdot\left(1+3^2+3^4\right)+...+3^{1987}\cdot\left(1+3^2+3^4\right)\)
\(\Rightarrow B=3\cdot91+...+3^{1987}\cdot91\)
\(\Rightarrow B=91\cdot\left(3+...+3^{1987}\right)\)
\(\Rightarrow13\cdot7\cdot\left(3+...+3^{1987}\right)⋮13\left(dpcm\right)\)
B = \(\frac{3}{3.6}+\frac{3}{6.9}+...+\frac{3}{53.56}\)
B = \(\frac{6-3}{3.6}+\frac{9-6}{6.9}+...+\frac{56-53}{53.56}\)
B = \(\frac{6}{3.6}-\frac{3}{3.6}+...+\frac{56}{53.56}-\frac{53}{53.56}\)
B = \(\frac{1}{3}-\frac{1}{6}+...+\frac{1}{53}-\frac{1}{56}\)
B = \(\frac{1}{3}-\frac{1}{56}\)
B = \(\frac{53}{168}\)
Ta có:
\(B=\frac{3}{3.6}+\frac{3}{6.9}+\frac{3}{9.11}+...+\frac{3}{53.56}\)
\(=\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}+...+\frac{1}{53}-\frac{1}{56}\)
\(=\frac{1}{3}-\frac{1}{56}=\frac{53}{168}\)
Vậy B=\(\frac{53}{168}\)
\(A=\frac{3}{1.3}+\frac{3}{3.5}+\frac{3}{5.7}+...+\frac{3}{49.51}\)
\(A=\frac{3}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{49.51}\right)\)
\(A=\frac{3}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{49}-\frac{1}{51}\right)\)
\(A=\frac{3}{2}.\left(1-\frac{1}{51}\right)\)
\(A=\frac{3}{2}.\frac{50}{51}=\frac{25}{17}\)
\(A=\frac{3}{1.3}+\frac{3}{3.5}+\frac{3}{5.7}+...+\frac{3}{49.51}\)
\(A=3.\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+....+\frac{1}{49}-\frac{1}{50}\right)\)
\(A=\frac{3}{2}\left(1-\frac{1}{50}\right)\)
\(A=\frac{3}{2}.\frac{49}{50}\)
\(A=\frac{147}{100}\)