2/1.3+2/3.5+2/5.7+…….+2/x(x+2)<2003/2004 20/08/2021 Bởi Claire 2/1.3+2/3.5+2/5.7+…….+2/x(x+2)<2003/2004
Đáp án: `x<2002` Giải thích các bước giải: `2/1.3+2/3.5+2/5.7+…+2/(x(x+2))<2003/2004` `1-1/3+1/3-1/5+1/5-1/7+…+1/x-1/(x+2)<2003/2004` `1-1/(x+2)<2003/2004` `1-1/(x+2)<1-1/2004` Vì `1-1/(x+2)<1-1/2004` nên `1/(x+2)>1/2004` `x+2<2004` `x<2004-2` `x<2002` Bình luận
Giải thích các bước giải: Ta có:$A=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+…+\dfrac{2}{x(x+2)}$$\to A=\dfrac{3-1}{1.3}+\dfrac{5-3}{3.5}+\dfrac{7-5}{5.7}+…+\dfrac{x+2-x}{x(x+2)}$$\to A=\dfrac11-\dfrac13+\dfrac13-\dfrac15+\dfrac15-\dfrac17+…+\dfrac1{x}-\dfrac1{x+2}$$\to A=1-\dfrac1{x+2}$ Để $A<\dfrac{2003}{2004}$ $\to 1-\dfrac1{x+2}<\dfrac{2003}{2004}$ $\to \dfrac1{x+2}>1-\dfrac{2003}{2004}$ $\to \dfrac1{x+2}>\dfrac1{2004}$ $\to x+2<2004$ $\to x<2002$ Mà $x\in N^*\to x\in\{1, 2, 3, 4, .., 2001\}$ Bình luận
Đáp án:
`x<2002`
Giải thích các bước giải:
`2/1.3+2/3.5+2/5.7+…+2/(x(x+2))<2003/2004`
`1-1/3+1/3-1/5+1/5-1/7+…+1/x-1/(x+2)<2003/2004`
`1-1/(x+2)<2003/2004`
`1-1/(x+2)<1-1/2004`
Vì `1-1/(x+2)<1-1/2004` nên `1/(x+2)>1/2004`
`x+2<2004`
`x<2004-2`
`x<2002`
Giải thích các bước giải:
Ta có:
$A=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+…+\dfrac{2}{x(x+2)}$
$\to A=\dfrac{3-1}{1.3}+\dfrac{5-3}{3.5}+\dfrac{7-5}{5.7}+…+\dfrac{x+2-x}{x(x+2)}$
$\to A=\dfrac11-\dfrac13+\dfrac13-\dfrac15+\dfrac15-\dfrac17+…+\dfrac1{x}-\dfrac1{x+2}$
$\to A=1-\dfrac1{x+2}$
Để $A<\dfrac{2003}{2004}$
$\to 1-\dfrac1{x+2}<\dfrac{2003}{2004}$
$\to \dfrac1{x+2}>1-\dfrac{2003}{2004}$
$\to \dfrac1{x+2}>\dfrac1{2004}$
$\to x+2<2004$
$\to x<2002$
Mà $x\in N^*\to x\in\{1, 2, 3, 4, .., 2001\}$