Tính bằng cách thuận thiện:
\(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{100}}\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\left(1+\dfrac{1}{1.3}\right)\left(1+\dfrac{1}{2.4}\right)...\left(1+\dfrac{1}{99.101}\right)\)
\(=\dfrac{2^2}{1.3}.\dfrac{3^2}{2.4}.\dfrac{4^2}{3.5}....\dfrac{100^2}{99.101}\)
\(=\dfrac{2.3.4...100}{1.2.3.4...99}.\dfrac{2.3.4...100}{3.4.5....101}\)
\(=\dfrac{100}{1}.\dfrac{2}{101}\)
\(=\dfrac{200}{101}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Bài 3
a,26/100+0,009+41/100+0,24
0,26+0,09+0,41+0,24
(0,26+0,24)+(0,09+0,41)
0,5+0,5
=1
b,9+1/4+6+2/7+7+3/5+8+2/3+2/5+1/3+5/7+3/4
(9+6+7+8)+(2/7+5/7)+(1/4+3/4)+(3/5+2/5)+(2/3+1/3)
30+1+1+1+1
=34
Bài 4,5 khó quá mik ko bít lamf^^))
Bài 4: a, \(\dfrac{2008}{2009}\) < 1; \(\dfrac{10}{9}\) > 1
\(\dfrac{2008}{2009}\) < \(\dfrac{10}{9}\)
b, \(\dfrac{1}{a+1}\) và \(\dfrac{1}{a-1}\)
Ta có: a + 1 > a - 1 ⇒ \(\dfrac{1}{a+1}\) < \(\dfrac{1}{a-1}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a)\(=\left(\dfrac{2}{2}+\dfrac{1}{2}\right)\times\left(\dfrac{3}{3}+\dfrac{1}{3}\right)\times...\times\left(\dfrac{2005}{2005}+\dfrac{1}{2005}\right)\)
\(=\dfrac{3}{2}\times\dfrac{4}{3}\times...\times\dfrac{2006}{2005}=\dfrac{2006}{2}=1003\)
b)\(=\left(\dfrac{2}{3}+\dfrac{1}{3}\right)\times\dfrac{1}{2}=\dfrac{3}{3}\times\dfrac{1}{2}=\dfrac{1}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
( 1 + 1/2 ) . ( 1+ 1/3) . ( 1+ 1/4 ) . ( 1+ 1/5 )
=3/2 . 4/3.5/4.6/5
= 3.4.5.6/2.3.4.5
=6/2 = 3
= 1 x ( \(\dfrac{1}{2}\) + \(\dfrac{1}{3}\) + \(\dfrac{1}{4}\) + \(\dfrac{1}{5}\) )
= 1 x \(\dfrac{77}{60}\)
= \(\dfrac{77}{60}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(B=\dfrac{1+\dfrac{1}{7}+\dfrac{1}{7^2}-\dfrac{1}{7^3}}{4+\dfrac{4}{7}+\dfrac{4}{7^2}-\dfrac{4}{7^3}}\cdot\dfrac{858585}{313131}\cdot\left(-1\dfrac{14}{17}\right)\)
\(=\dfrac{1}{4}\cdot\dfrac{85}{31}\cdot\dfrac{-31}{17}\)
\(=\dfrac{-5}{4}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a, \(=\dfrac{1+4}{5}+\dfrac{5+1+3}{9}=1+1=2\)
b, \(=\dfrac{1+4+2}{3}+\dfrac{1+2+5}{6}=\dfrac{6}{3}+\dfrac{8}{6}=2+\dfrac{4}{3}=\dfrac{6+4}{3}=\dfrac{10}{3}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a: =20/90+15/90+54/90=89/90
b: =(9/13+4/13)+2/3=1+2/3=5/3
A = 1 + \(\dfrac{1}{2}\) + \(\dfrac{1}{2^2}\)+......+\(\dfrac{1}{2^{99}}\)+ \(\dfrac{1}{2^{100}}\)
2A = 2 + 1 + \(\dfrac{1}{2}\) + \(\dfrac{1}{2^2}\)+.....+ \(\dfrac{1}{2^{99}}\)
2A - A = 2 - \(\dfrac{1}{2^{100}}\)
A = 2 - \(\dfrac{1}{2^{100}}\)
\(A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{100}}\\ \Rightarrow\dfrac{1}{2}A=\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{101}}\)
\(\Rightarrow A-\dfrac{1}{2}A=1-\dfrac{1}{2^{101}}\\ \Rightarrow\dfrac{1}{2}A=\dfrac{2^{101}-1}{2^{101}}\\ \Rightarrow A=\dfrac{2^{201}-1}{2^{100}}\)