a) Ta có : \(\dfrac{a}{2}=\dfrac{b}{3}=\dfrac{c}{4}\)

\(\Rightarrow\dfrac{a}{2}=\dfrac{2b}{6}=\dfrac{3c}{12}=\dfrac{a+2b+3c}{2+6+12}=\dfrac{-20}{20}=-1\)

\(\Rightarrow\left\{{}\begin{matrix}a=\left(-1\right)\cdot2=-2\\b=\dfrac{\left(-1\right).6}{2}=-3\\c=\dfrac{\left(-1\right).12}{3}=-4\end{matrix}\right.\)

b) Ta có : \(S=\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{9900}\)

\(=\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100}\)

\(=1-\dfrac{1}{100}=\dfrac{99}{100}\).

Vậy : \(S=\dfrac{99}{100}.\)

a)\(\dfrac{a}{2}=\dfrac{2b}{6}=\dfrac{3c}{12}=\dfrac{a+2b+3c}{2+6+12}=-\dfrac{20}{20}=-1\)

\(\left\{{}\begin{matrix}\dfrac{a}{2}=-1\Leftrightarrow a=-2\\\dfrac{b}{3}=-1\Leftrightarrow b=-3\\\dfrac{c}{4}=-1\Leftrightarrow c=-4\end{matrix}\right.\)

b)\(S=\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{9900}\\ =\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}\\ =\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{99}-\dfrac{1}{100}\\ =1-\dfrac{1}{100}=\dfrac{99}{100}\)

2.Giải:

Theo bài ra ta có:

\(\frac{a}{2}=\frac{b}{3}=\frac{c}{4}=\frac{d}{5}\) và a + b + c + d = -42

Theo tính chất dãy tỉ số bằng nhau ta có:
\(\frac{a}{2}=\frac{b}{3}=\frac{c}{4}=\frac{d}{5}=\frac{a+b+c+d}{2+3+4+5}=\frac{-42}{14}=-3\)

+) \(\frac{a}{2}=-3\Rightarrow a=-6\)

+) \(\frac{b}{3}=-3\Rightarrow b=-9\)

+) \(\frac{c}{4}=-3\Rightarrow c=-12\)

+) \(\frac{d}{5}=-3\Rightarrow d=-15\)

Vậy a = -6

        b = -9

        c = -12

        d = -15

Bài 3:

Ta có:\(\frac{a}{2}=\frac{b}{3}\Leftrightarrow\frac{a}{10}=\frac{b}{15}\); \(\frac{b}{5}=\frac{c}{4}\Leftrightarrow\frac{b}{15}=\frac{c}{12}\)

\(\Rightarrow\frac{a}{10}=\frac{b}{15}=\frac{c}{12}\)

Áp dụng tc dãy tỉ:

\(\frac{a}{10}=\frac{b}{15}=\frac{c}{20}=\frac{a+b+c}{10+15+12}=\frac{-49}{37}\)

Với \(\frac{a}{10}=\frac{-49}{37}\Rightarrow a=10\cdot\frac{-49}{37}=\frac{-490}{37}\)

Với \(\frac{b}{15}=\frac{-49}{37}\Rightarrow b=15\cdot\frac{-49}{37}=\frac{-735}{37}\)

Với \(\frac{c}{12}=\frac{-49}{37}\Rightarrow c=12\cdot\frac{-49}{37}=\frac{-588}{37}\)

Bài 2:

a : b : c : d = 2 : 3 : 4 : 5 \(\Rightarrow\frac{a}{2}=\frac{b}{3}=\frac{c}{4}=\frac{d}{5}\)

Áp dụng tc dãy tỉ:

\(\frac{a}{2}=\frac{b}{3}=\frac{c}{4}=\frac{d}{5}=\frac{a+b+c+d}{2+3+4+5}=\frac{-42}{14}=-3\)

Với \(\frac{a}{2}=-3\Rightarrow a=-6\)

Với \(\frac{b}{3}=-6\Rightarrow b=-18\)

Với \(\frac{c}{4}=-6\Rightarrow c=-24\)

Với \(\frac{d}{5}=-6\Rightarrow d=-30\)

Theo đề, ta có: 

\(\frac{a}{2}=\frac{b}{3}=\frac{c}{4}\) và a+2b-3c=-20

Áp dụng tính chất của dãy tỉ số bằng nhau ta có:

\(\frac{a}{2}=\frac{b}{3}=\frac{c}{4}=\)\(\frac{a}{2}=\frac{2b}{6}=\frac{3c}{12}=\frac{a+2b-3c}{2+6-12}=\frac{-20}{-4}=5\)

=> \(\frac{a}{2}=5\)

\(\frac{b}{3}=5\)

\(\frac{c}{4}=5\)

=> a =10

b =15

c =20

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\(\dfrac{a}{2}=\dfrac{b}{3}=\dfrac{c}{4}=\dfrac{a+2b-3c}{2+2\cdot3-3\cdot4}=\dfrac{-20}{-4}=5\\ \Rightarrow\left\{{}\begin{matrix}a=10\\b=15\\c=20\end{matrix}\right.\)

\(\dfrac{a}{2}=\dfrac{b}{3}=\dfrac{c}{4}=\dfrac{2b}{6}=\dfrac{3c}{12}=\dfrac{a+2b-3c}{2+6-12}=\dfrac{-20}{-4}=5\)

\(\Rightarrow\left\{{}\begin{matrix}\dfrac{a}{2}=5\\\dfrac{b}{3}=5\\\dfrac{c}{4}=5\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=10\\b=15\\c=20\end{matrix}\right.\)

`a/2 = b/3 = c/4`

`=> a/2 = (2b)/6 = (3c)/12`

mà `a+2b-3c=-20`

áp dụng tính chất dãy tỉ số bằng nhau ta có

` a/2 = (2b)/6 = (3c)/12 = (a+2b-3c)/(2+6-12)=(-20)/-4 = 5`

` => a=5xx2=10`

`b=5xx3=15`

`c=5xx4=20`

ta có : `a/2=b/3=c/4 =>a/2=(2b)/6=(3c)/12` và `a+2b-3c=-20`

ADTC dãy tỉ số bằng nhau ta có :

`a/2=(2b)/6=(3c)/6=(a+2b-3c)/(2+6-12)=(-20)/-4=5`

`=>a/2=5=>a=5.2=10`

`=>b/3=5=>b=5.3=15`

`=>c/4=5=>c=5.4=20`

#\(N\)

`a/2 = b/3 = c/4 , a+2b-3c = -20`

`-> a/2 =`\(\dfrac{2b}{6}=\dfrac{3c}{12}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{a}{2}=\dfrac{2b}{6}=\dfrac{3c}{12}=\dfrac{a+2b-3c}{2+6-12}=\dfrac{-20}{-4}=5\)

`-> a/2 = 5 , b/3 = 5 , c/4 = 5`

`-> a=2.5 = 10 , b=3.5=15 , c=4.5=20`