Question

Tính tổng S=3+3/2+3/2²+...+3/2⁹

Answer 1

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

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

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

\(\Rightarrow2N-N=\left(2+1+\frac{1}{2}+...+\frac{1}{2^8}\right)-\left(1+\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^9}\right)\)

\(\Rightarrow N=2-\frac{1}{2^9}\)

Khi đó \(S=3.N=3.\left(2-\frac{1}{2^9}\right)=6-\frac{3}{2^9}=\frac{3069}{512}\)