\(S=3+\frac{3}{2}+\frac{3}{2^2}+.....+\frac{3}{2^9}\)

\(\Rightarrow\frac{1}{2}S=\frac{3}{2}+\frac{3}{2^2}+\frac{3}{2^3}+.....+\frac{3}{2^{10}}\)

\(\Rightarrow S-\frac{1}{2}S=\left(3+\frac{3}{2}+\frac{3}{2^2}+....+\frac{3}{3^9}\right)-\left(\frac{3}{2}+\frac{3}{2^2}+.....+\frac{3}{2^{10}}\right)\)

\(\Rightarrow\frac{S}{2}=3-\frac{3}{2^{10}}\)

\(\Rightarrow S=\left(3-\frac{3}{2^{10}}\right).2\)\(=6-\frac{3}{2^9}\)

\(S=3\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^9}\right)\)

Đặt \(A=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^9}\)

\(\Rightarrow2A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^8}\)

\(\Rightarrow2A-A=A=1-\frac{1}{2^9}\)

Do đó \(S=3\left(1-\frac{1}{2^9}\right)=3\left(1-\frac{1}{512}\right)=3-\frac{3}{512}=\frac{1533}{512}\)

S= 3+\(\frac{3}{2}\)+\(\frac{3}{2^2}\)+......+\(\frac{3}{2^9}\)

S= 3(1+\(\frac{1}{2}\)+\(\frac{1}{2^2}\)+........\(\frac{1}{2^9}\))

\(\frac{S}{3}\)= 1+\(\frac{1}{2}\)+\(\frac{1}{2^2}\)+......+\(\frac{1}{2^9}\)   (1)

\(\frac{2S}{3}\)=2+1+\(\frac{1}{2}\)+.............+\(\frac{1}{2^8}\)    (2)

trừ cả 2 vế của (1) và (2) ta được

\(\frac{S}{3}\)=2 -\(\frac{1}{2^9}\)

\(\frac{S}{3}\)=\(\frac{2^{10}-1}{2^9}\)

s=\(\frac{2^{10}-1}{3\cdot2^9}\)