Question

cho a=2/3+8/9+26/27+...+3^n-1/3^n

CM a>n-1/2

Answer 1

Lời giải:
\(A=1-\frac{1}{3}+1-\frac{1}{9}+1-\frac{1}{27}+...+1-\frac{1}{3^n}\\ =n-(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^n})\)

\(=n-M\)

Xét $M=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+....+\frac{1}{3^n}$

$3M=1+\frac{1}{3}+\frac{1}{3^2}+....+\frac{1}{3^{n-1}}$

$\Rightarrow 3M-M = 1-\frac{1}{3^n}$

$\Rightarrow M=\frac{1}{2}-\frac{1}{2.3^n}< \frac{1}{2}$

$\Rightarrow A=n-M> n-\frac{1}{2}$