Chứng tỏ rằng: 1/2*3+1/3*4+1/4*5+....+1/99*100<1/2
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PT
0
NP
0
TN
1
YN
19 tháng 5 2021
* Bỏ ngoặc vuông đi :(
\(\text{Ta có:}\)
\(200-\left(3+\frac{2}{3}+\frac{2}{4}+...+\frac{2}{100}\right)\)
\(\rightarrow200-2-\left(1+\frac{2}{3}+...+\frac{2}{100}\right)\)
\(\rightarrow198-\left(1+\frac{2}{3}+...+\frac{2}{100}\right)\)
\(\rightarrow198-\left(1+\frac{2}{3}+...+\frac{2}{100}\right)\)
\(\rightarrow2.[99-\left(\frac{1}{2}-\frac{1}{3}+...+\frac{1}{100}\right)]\) \(\left(1\right)\)
\(\text{Ta có:}\)
\(\frac{1}{2}+\frac{2}{3}+...+\frac{99}{100}\)
\(\text{Rút}\)\(\left(1\right)\)\(\text{ra có 99 số}\)
\(\rightarrow99-\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}\right)\) \(\left(2\right)\)
\(\text{Từ}\)\(\left(1\right)\)\(\text{và}\)\(\left(2\right)\)\(\Rightarrow\)\(200-\left(3+\frac{2}{3}+\frac{2}{4}+\frac{2}{5}+...+\frac{2}{100}\right):\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{99}{100}\right)=2\)
NK
2
XO
3 tháng 6 2019
Ta có : \(\frac{1}{2}+\frac{2}{3}+..+\frac{99}{100}\)
= \((1-\frac{1}{2})+(1-\frac{1}{3})+...+(1-\frac{99}{100})\)(100 cặp số )
= \(\left(1+1+1+...+1\right)-\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}\right)\)(100 số hạng 1)
= \(1\times100-\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+..+\frac{1}{100}\right)\)
= \(100-\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}\right)\)
=> 100-(1+1/2+1/3+...+1/100) = 1/2+2/3+3/4+...+99/100
NN
1
NV
Nguyễn Việt Lâm
Giáo viên
21 tháng 4 2021
\(2^2< 2.3\Rightarrow\dfrac{1}{2^2}>\dfrac{1}{2.3}=\dfrac{1}{2}-\dfrac{1}{3}\)
Tương tự: \(\dfrac{1}{3^2}>\dfrac{1}{3}-\dfrac{1}{4}\) ; \(\dfrac{1}{4^2}>\dfrac{1}{4}-\dfrac{1}{5}\) ; ....; \(\dfrac{1}{100^2}>\dfrac{1}{100}-\dfrac{1}{101}\)
Do đó:
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{100^2}>\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{100}-\dfrac{1}{101}\)
\(\Leftrightarrow\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{100^2}>\dfrac{1}{2}-\dfrac{1}{101}\)
\(\Leftrightarrow\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{100^2}>\dfrac{99}{202}\)
CN
2
TN
21 tháng 5 2016
= (1+1/3+1/5+…+1/99)-(1/2+1/4+….+1/100)
= (1+1/2+1/3+…+1/100)-2(1/2+1/4+1/6+…+1/100)
= (1+1/2+1/3+…+1/100)-(1+1/2+1/3+…+1/50)
=1/51+1/52+…+1/100=VP (đpcm)
21 tháng 5 2016
= (1+1/3+1/5+…+1/99)-(1/2+1/4+….+1/100)
= (1+1/2+1/3+…+1/100)-2(1/2+1/4+1/6+…+1/100)
= (1+1/2+1/3+…+1/100)-(1+1/2+1/3+…+1/50)
=1/51+1/52+…+1/100=VP (đpcm)
\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}< \frac{1}{2}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}< \frac{1}{2}\)
\(=\frac{1}{2}-\frac{1}{100}< \frac{1}{2}\left(đpcm\right)\)