\(S=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{2187}\)
\(=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^7}\)
\(\Rightarrow\)\(3S=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^6}\)
\(\Rightarrow\)\(3S-S=\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^6}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^7}\right)\)
\(\Rightarrow\)\(2S=1-\frac{1}{3^7}\)
\(\Rightarrow\)\(S=\frac{1-\frac{1}{3^7}}{2}\)
\(S=\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^7}\)
\(3S=1+\frac{1}{3}+...+\frac{1}{3^6}\)
\(3S-S=\left(1+\frac{1}{3}+...+\frac{1}{3^6}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^7}\right)\)
\(2S=1-\frac{1}{3^7}\)
\(S=\frac{1-\frac{1}{3^7}}{2}\)
\(S=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{2187}\)
\(S=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^7}\)
\(\Rightarrow3S=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^6}\)
\(\Rightarrow3S-S=1-\frac{1}{3^7}\)
\(2S=1-\frac{1}{3^7}\)
\(\Rightarrow S=\frac{1-\frac{1}{3^7}}{2}\)
\(S=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{2187}.\)
\(=\frac{1}{3^1}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+...+\frac{1}{3^7}\)[ BƯỚC NÀY BẠN BỎ CŨNG ĐƯỢC ]
\(=\frac{729}{2187}+\frac{81}{2187}+\frac{27}{2187}+\frac{9}{2187}+\frac{3}{2187}+\frac{1}{2781}\)
\(=\frac{850}{2187}\)
[ THAM KHẢO THÔI]