Ta có: S = \(\dfrac{1}{3}+\dfrac{3}{3.7}+\dfrac{5}{3.7.11}+...+\dfrac{2n+1}{3.7.11...\left(4n+3\right)}\)

⇒ 2S = \(\dfrac{2}{3}+\dfrac{6}{3.7}+\dfrac{10}{3.7.11}+...+\dfrac{4n+2}{3.7.11...\left(4n+3\right)}\)

⇒ 2S + \(\dfrac{1}{3.7.11...\left(4n+3\right)}\) = \(\dfrac{2}{3}+\dfrac{6}{3.7}+\dfrac{10}{3.7.11}+...+\dfrac{4n+3}{3.7.11...\left(4n+3\right)}\)

Đến đây nó sẽ rút gọn liên tục và sau nhiều lần rút gọn ta có:

2S + \(\dfrac{1}{3.7.11...\left(4n+3\right)}\) = \(\dfrac{2}{3}+\dfrac{6}{3.7}+\dfrac{10}{3.7.11}+\dfrac{1}{3.7.11}\) = \(\dfrac{2}{3}+\dfrac{6}{3.7}+\dfrac{11}{3.7.11}\) = \(\dfrac{2}{3}+\dfrac{6}{3.7}+\dfrac{1}{3.7}\) = \(\dfrac{2}{3}+\dfrac{7}{3.7}=\dfrac{2}{3}+\dfrac{1}{3}=1\)

Suy ra 2S < 1 ⇒ S < \(\dfrac{1}{2}\)(đpcm)

8C=8/3.7.11+8/7.11.15+8/11.15.19+8/15.19.23+8/19.23.27+8/23.27.31

8C=1/3.7-1/7.11+1/7.11-1/11.15+1/11.15-1/15.19+1/15.19-1/19.23+1/19.23+-1/23.27-1/27.31

8C=1/3.7-1/27.31

8C=1/21-1/837

8C=272/5859

C=34/5859

a: \(=\dfrac{21\cdot11}{22\cdot9}=\dfrac{1}{2}\cdot\dfrac{7}{3}=\dfrac{7}{6}\)

b: \(=\dfrac{49\cdot8}{10}=49\cdot\dfrac{4}{5}=\dfrac{196}{5}\)

c: \(=\dfrac{12\cdot\left(-4\right)}{32\cdot6}=\dfrac{-48}{192}=-\dfrac{1}{4}\)

=21⋅1122⋅9=12⋅73=76

4x-(7+5x)=15+(-29).                      b,x/11=x+4/33

4x-7-5x=-14.                                    33x=11(x+4)

4x-5x=14+7.                                     33x=11x+44

-x=21.                                               33x-11x=44

x=-21                                                22x=44

                                                         x=44:22=2

2.     a,   \(\frac{3.7.11}{33.9}\)

=         \(\frac{3.7.11}{3.11.3.3}\)

=          \(\frac{7}{9}\)

b,             \(\frac{5^3.2^4}{4.10^2.11}\)

=              \(\frac{5^3.2^4}{2^2.5^2.2^2.11}\)

=              \(\frac{5.1}{11}=\frac{5}{11}\)

Trl

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#Mưaa

\(\dfrac{-63}{72}=\dfrac{-7}{8}\)

\(\dfrac{20}{-140}=\dfrac{-1}{7}\)

\(\dfrac{3.10}{5.24}=\dfrac{30}{120}=\dfrac{1}{4}\)

\(\dfrac{3.7.11}{22.9}=\dfrac{231}{198}=\dfrac{7}{6}\)

\(\dfrac{8.5-8.2}{16}=\dfrac{24}{16}=\dfrac{3}{2}\)

\(\dfrac{11.4-11}{2-13}=\dfrac{33}{-11}=-3\)

Bài 1: 

a: =-3/7

b: =-3/5

c: =1/13

d: =-1/2

Bài 2: 

a: \(=\dfrac{21\cdot11}{22\cdot9}=\dfrac{1}{2}\cdot\dfrac{7}{3}=\dfrac{7}{6}\)

b: \(=\dfrac{49\cdot8}{10}=\dfrac{196}{5}\)

\(\frac{3.7.11}{44.12}=\frac{3.7.11}{4.11.3.4}=\frac{7}{4.4}=\frac{7}{16}\)

\(\frac{22+77}{7.5.11}=\frac{99}{7.5.11}=\frac{9.11}{7.5.11}=\frac{9}{35}\)

\(\frac{3.7.11}{44.12}=\frac{3.7.11}{11.4.3.4}=\frac{7}{4^2}\)

\(\frac{22+77}{7.5.11}=\frac{11\left(2+7\right)}{35.11}\frac{9}{35}\)

\(a=\frac{3.7.11}{44.12}=\frac{231}{528}=\frac{7}{16}\\ b=\frac{22+77}{7.5.11}=\frac{99}{385}=\frac{9}{35}\)