Question

Cho C = 2/3 + 8/9 + 26/27 + ... + 3^n-1/3^n

Chứng ming rằng : C > n - 1/2

Answer 1

\(C=\frac{3-1}{3}+\frac{3^2-1}{3^2}+...+\frac{3^n-1}{3^n}\)

\(=1-\frac{1}{3}+1-\frac{1}{3^2}+...+1-\frac{1}{3^n}\)

\(=1+1+...+1-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^n}\right)\)

\(=n-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^n}\right)=n-D\)

\(D=\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^n}\)

\(3D=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{n-1}}\)

\(\Rightarrow2D=1-\frac{1}{3^n}\Rightarrow D=\frac{1}{2}-\frac{1}{2.3^n}\)

\(\Rightarrow C=n-\left(\frac{1}{2}-\frac{1}{2.3^n}\right)=n-\frac{1}{2}+\frac{1}{2.3^n}>n-\frac{1}{2}\)