\(A=1+\frac{2}{6}+\frac{2}{12}+...+\frac{2}{380}\)
\(=1+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{19.20}\)
\(=1+2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{19}-\frac{1}{20}\right)\)
\(=1+2\left(\frac{1}{2}-\frac{1}{20}\right)\)
\(=1+2\times\frac{9}{20}\)
\(=1+\frac{9}{10}\)
\(=\frac{19}{10}\)
b)\(2S=2\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{20}}\right)\)
\(2S=1+\frac{1}{2}+...+\frac{1}{2^{19}}\)
\(2S-S=\left(1+\frac{1}{2}+...+\frac{1}{2^{19}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{20}}\right)\)
\(S=1-\frac{1}{2^{20}}\)
c)đặt A=1+2+2^2+2^3+...+2^2006+2^2007.
2A=2(1+2+2^2+2^3+...+2^2006+2^2007)
2A=2+2^2+2^3+...+2^2008
2A-A=(2+2^2+2^3+...+2^2008)-(1+2+2^2+2^3+...+2^2006+2^2007)
A=2^2008-1
