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\(S=\frac{101}{120}+\frac{1}{2.3}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...\frac{1}{18.19}+\frac{1}{19.20}\right)\)
\(S=\frac{101}{120}+\frac{1}{6}\left(\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{19-18}{18.19}+\frac{20-19}{19.20}\right)\)
\(S=\frac{101}{120}+\frac{1}{6}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{18}-\frac{1}{19}+\frac{1}{19}-\frac{1}{20}\right)\)
\(S=\frac{101}{120}+\frac{1}{6}\left(1-\frac{1}{20}\right)=\frac{101}{120}+\frac{19}{120}=\frac{120}{120}=1\)
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= 2.(1 / 2.3 + 1 / 3.4 + ..... + 1 / x (x + 1) = 2007/2009
= 2.(1/2 - 1/3 + 1/3 - +.......+ 1/x - 1/x+1) = 2007/2009
= 2.( 1/2 - 1/x+1) = 2007/2009
= 1 - 1/x+1 =2007/2009
= 1/x+1 = 1/2009
=> x + 1 = 2009
=> x = 2008
Ta có: 2/2.3 + 2/3.4 + .... + 2/x.(x+1) = 2007/2009
=> 2.[1/2.3+1/3.4+.....+1/x.(x+1)]=2007/2009
=> 2.(1/2-1/3+1/3-1/4 + .... + 1/x - 1/x+1) = 2007/2009
=> 2.(1/2-1/x+1)=2007/2009
=>1/2 - 1/x+1 = 2007/2009 : 2
=> 1/2 - 1/x+1 = 2007/4018
=> 1/x+1 = 2007/4018 +1/2
=> 1/x+1 =
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A= 1/4 ( 1/2x6 +1/6x10 +.............+1/46x50)
A= 1/4 ( 1/2 - 1/6 + 1/6 - 1/10 +.......... + 1/46 - 1/50 )
A= 1/4 ( 1/2 - 1/50 )
A= 1/4 x 12/25
A= 3/25
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Ta có : S1 = 1 + (-3) + 5 + (-7) + .... + 17
= (1 - 3) + (5 - 7) + (9 - 11)+ (13 - 15) + 17
= -2 + -2 + -2 + -2 + 17
= -2 x 4 + 17
= -8 + 17
S1 = 9
S2 = (4 - 2) + (8 - 6) + (12 - 10) + (16 - 14) + -18
= 2 x 4 - 18
S2 = -10
S1 + S2 = 9 - 10 = -1
S1=1+(-3)+5+(-7)+...+17.
S1=-2+(-2)+....+(-2).(9 số -2).
S2=-2+4+(-6)+....+(-18)
S2=-2+(-2)+...+(-2).(9 số -2).
=> (-2).(9+9)=-36.
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ta có:\(\dfrac{101^{120}+1}{101^{103}+1}>1;\dfrac{101^{103}+1}{101^{104}+1}< 1\) => N<1<M
vậy N<M
Ta có M=\(\frac{101^{120}+1}{101^{103}+1}>1\)
N=\(\frac{101^{103}+1}{101^{104}+1}< 1\)
=>\(\frac{101^{103}+1}{101^{104}+1}< 1< \frac{101^{120}+1}{101^{103}+1}\)
=>\(\frac{101^{103}+1}{101^{104}+1}< \frac{101^{120}+1}{101^{103}+1}\)
=> N < M
Vaayj N < M
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\(\dfrac{1}{1.3}+\dfrac{1}{2.3}+\dfrac{1}{2.5}+\dfrac{1}{3.5}+\dfrac{1}{3.7}+\dfrac{1}{4.7}+\dfrac{1}{4.9}\)
\(=\dfrac{1}{1.3}+\dfrac{1}{3.2}+\dfrac{1}{2.5}+\dfrac{1}{5.3}+\dfrac{1}{3.7}+\dfrac{1}{7.4}+\dfrac{1}{4.9}\)
\(=\left(\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{6.7}+\dfrac{1}{7.8}+\dfrac{1}{8.9}\right):\dfrac{1}{2}\)
\(=\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{9}\right):\dfrac{1}{2}\)
\(=\left(\dfrac{1}{2}-\dfrac{1}{9}\right):\dfrac{1}{2}\)
\(=\dfrac{7}{18}:\dfrac{1}{2}\)
\(=\dfrac{7}{9}\)