Question

Cho A =\(\dfrac{1}{2^2}\)+\(\dfrac{1}{2^4}\)+\(\dfrac{1}{2^6}\)+....+\(\dfrac{1}{2^{100}}\).C/minh A <\(\dfrac{1}{3}\)

Answer 1

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

\(2^2A=2^2\left(\dfrac{1}{2^2}+\dfrac{1}{2^4}+\dfrac{1}{2^6}+...+\dfrac{1}{2^{100}}\right)\)

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

\(4A-A=\left(1+\dfrac{1}{2^2}+\dfrac{1}{2^4}+...+\dfrac{1}{2^{98}}\right)-\left(\dfrac{1}{2^2}+\dfrac{1}{2^4}+\dfrac{1}{2^6}+...+\dfrac{1}{2^{100}}\right)\)\(3A=1-\dfrac{1}{2^{100}}\)

\(A=\dfrac{1-\dfrac{1}{2^{100}}}{3}\)

\(A=\dfrac{1}{3}-\dfrac{\dfrac{1}{2^{100}}}{3}< \dfrac{1}{3}\)

Vậy \(A< \dfrac{1}{3}\)