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\(3A=5+\dfrac{5}{3}+\dfrac{5}{3^2}+...+\dfrac{5}{3^{19}}\)
\(2A=3A-A=5-\dfrac{5}{3^{20}}\)
\(A=\dfrac{5.3^{20}-5}{2.3^{20}}\)
\(\dfrac{5}{3}+\dfrac{5}{3^2}+...+\dfrac{5}{3^{20}}=5\left(\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{20}}\right)\)
Gọi \(A=\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{20}}\). Ta có
\(3A=1+\dfrac{1}{3}+...+\dfrac{1}{3^{19}}\)
\(3A-A=\left(1+\dfrac{1}{3}+...+\dfrac{1}{3^{19}}\right)-\left(\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{20}}\right)\)
\(2A=1-\dfrac{1}{3^{20}}\)
\(A=\dfrac{1-\dfrac{1}{3^{20}}}{2}\)
Suy ra \(\dfrac{5}{3}+\dfrac{5}{3^2}+...+\dfrac{5}{3^{20}}=5\left(\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{20}}\right)=5\cdot\dfrac{1-\dfrac{1}{3^{20}}}{2}=\dfrac{5-\dfrac{5}{3^{20}}}{2}\)
help mik với
