Question

[Trà My Phạm]

Cho A = 1/1+3 + 1/1+3+5 + 1/1+3+5+7 + 1/1+3+.....+2013

Chứng minh A < 3/4

Answer 1

[Nguyễn Thanh Hằng]

Ta có :

\(A=\dfrac{1}{1+3}+\dfrac{1}{1+3+5}+...........+\dfrac{1}{1+3+.....+2013}\)

\(A=\dfrac{1}{\dfrac{\left(1+3\right).2}{2}}+\dfrac{1}{\dfrac{\left(1+5\right).3}{2}}+.........+\dfrac{1}{\dfrac{\left(1+2013\right).1007}{2}}\)

\(A=\dfrac{2}{2.4}+\dfrac{2}{3.6}+\dfrac{2}{4.8}+...........+\dfrac{2}{1007.2014}\)

\(A=\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+..........+\dfrac{1}{1007.1007}\)

\(\Rightarrow A< \dfrac{1}{2.2}+\left(\dfrac{1}{2.3}+\dfrac{1}{3.4}+......+\dfrac{1}{1006.1008}\right)\)

\(\Rightarrow A< \dfrac{1}{4}+\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...........+\dfrac{1}{1006}-\dfrac{1}{1007}\right)\)

\(\Rightarrow A< \dfrac{1}{4}+\left(\dfrac{1}{2}-\dfrac{1}{1007}\right)\)

\(\Rightarrow A< \dfrac{1}{4}+\dfrac{1}{2}=\dfrac{3}{4}\) \(\rightarrowđpcm\)

~ Chúc bn học tốt ~