1/2x2/3x4/5x5/6x6/7x.....x98/99x99/100
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
1 + 2 x 2 + 3 x 3 + 4 x 4 + 5 x 5 + 6 x 6 + 7 x 7 + 8 x 8 + 9 x 9 + 10 x 10
= 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100
= 385
1 + 2x2 + 3x3 + 4x4 + 5x5 + 6x6 + 7x7 + 8x8 + 9x9 + 10x10
= 1+4+9+16+25+36+49+64+81+100
=(81+9)+(64+16)+(49+1)+)36+4)+25+100
=90+80+50+40+25 +100
=385
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
đặt \(A=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+\frac{1}{5.5}+\frac{1}{6.6}+\frac{1}{7.7}+\frac{1}{8.8}=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+\frac{1}{8^2}\)
\(A
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
đặt A=1/5x5 +1/6x6 + 1/7x7 + .....+ 1/100x100
=>A>1/5x6 + 1/6x7 +1/7x8 + .... + 1/100x101
=>A>1/5 - 1/6 + 1/6 - 1/7 + +1/7 - 1/8 + ..... + 1/100 - 1/101
=>A> 1/5 - 1/101
=>A>96/505 > 96/576 = 1/6
=>A>1/6
=>A>B
a>1/5x6+1/6x7+...+1/100x101
=1/5-1/6+1/6-1/7+...+1/100-1/101
=1/5-1/101
=101/505-5/101
=96/101
vì 96/101>1/6 nên a>1/6
![](https://rs.olm.vn/images/avt/0.png?1311)
a, \(\dfrac{20}{15}=\dfrac{4}{3}\)
b, \(\dfrac{6}{15}=\dfrac{2}{5}\)
c, \(\dfrac{42}{30}=\dfrac{7}{5}\)
d, \(\dfrac{40}{35}=\dfrac{8}{7}\)
e, \(\dfrac{42}{42}=1\)
g, \(\dfrac{54}{10}=\dfrac{27}{5}\)
h, \(\dfrac{12}{15}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49
8 x 8 = 64
9 x 9 = 81
10 x 10 = 100
\(1x1=1\)
\(2x2=4\)
\(3x3=9\)
\(4x4=16\)
\(5x5=25\)
\(6x6=36\)
\(7x7=49\)
\(8x8=64\)
\(9x9=81\)
\(10x10=100\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a)=1x2x3x4x5/2x3x4x5x6 =1/6
b)=1x2x3x4x5x6x7x8x9/10x9x8x7x6x5x4x3 x2=1/10
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta thấy:
1/2*2<1/1*2)vì 2*2>1*2).
1/3*3<1/2*3(vì 3*3>2*3).
...
1/8*8<1/7*8(vì 8*8>7*8).
=>1/2*2+1/3*3+1/4*4+...+1/8*8<1/1*2+1/2*3+1/3*4+...+1/7*8.
=>B<1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8.
=>B<1-1/8.
=>B<7/8.
Mà 7/8<1.
=>B<1.
Vậy B<1(đpcm).
`#3107.101107`
\(\dfrac{1}{2}\times\dfrac{2}{3}\times\dfrac{4}{5}\times\dfrac{5}{6}\times\dfrac{6}{7}\times...\times\dfrac{98}{99}\times\dfrac{99}{100}\)
\(=\left(\dfrac{1}{2}\times\dfrac{2}{3}\right)\times\dfrac{4}{5}\times\dfrac{5}{6}\times\dfrac{6}{7}\times...\times\dfrac{98}{99}\times\dfrac{99}{100}\)
\(=\dfrac{1}{3}\times\dfrac{4}{100}\)
\(=\dfrac{4}{300}\\ =\dfrac{1}{75}\)
\(\dfrac{1}{2}x\dfrac{2}{3}x\dfrac{3}{4}x\dfrac{4}{5}x\dfrac{5}{6}x\dfrac{6}{7}...x\dfrac{98}{99}x\dfrac{99}{100}\\ =\dfrac{1x2x3x4x5x6x...x98x99}{2x3x4x5x6x7x...x99x100}\\ =\dfrac{1}{100}\)