Question

Cho A=1/2^2+1/2^4+1/2^6+1/2^8+...+1/2^100

Chứng minh rằng A<1/3

Answer 1

\(A=\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}+\frac{1}{2^8}+...+\frac{1}{2^{100}}\)

\(2^2.A=1+\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}+...+\frac{1}{2^{98}}\)

\(2^2.A-A=\left(1+\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}+...+\frac{1}{2^{98}}\right)-\left(\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}+\frac{1}{2^8}+...+\frac{1}{2^{100}}\right)\)

\(4.A-A=1-\frac{1}{2^{100}}< 1\)

\(3A< 1\)

\(\Rightarrow A< \frac{1}{3}\left(đpcm\right)\)