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Với mọi x ta có :
+) \(\left|x+\dfrac{1}{1.3}\right|\ge0; \)
+) \(\left|x+\dfrac{1}{3.5}\right|\ge0;\)
.....................................
+) \(\left|x+\dfrac{1}{97.99}\right|\ge0\)
\(\Leftrightarrow\left|x+\dfrac{1}{1.3}\right|+\left|x+\dfrac{1}{3.5}\right|+.......+\left|x+\dfrac{1}{97.99}\right|\ge0\)
\(\Leftrightarrow50x\ge0\)
\(\Leftrightarrow x\ge0\)
Khi \(x\ge0\) ta được :
+) \(\left|x+\dfrac{1}{1.3}\right|=x+\dfrac{1}{1.3}\)
+) \(\left|x+\dfrac{1}{3.5}\right|=x+\dfrac{1}{3.5}\)
.............................................
+) \(\left|x+\dfrac{1}{97.99}\right|=x+\dfrac{1}{97.99}\)
\(\Leftrightarrow\left(x+\dfrac{1}{1.3}\right)+\left(x+\dfrac{1}{3.5}\right)+......+\left(x+\dfrac{1}{97.99}\right)=50x\)
\(\Leftrightarrow49x+\left(\dfrac{1}{1.3}+\dfrac{1}{3.5}+....+\dfrac{1}{97.99}\right)=50x\)
\(\Leftrightarrow x=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+....+\dfrac{1}{97}-\dfrac{1}{99}\)
\(\Leftrightarrow x=\dfrac{16}{99}\)
Vậy...
A\(A=\frac{1}{1.3}+..+\frac{1}{x\left(x+1\right)}\)
\(2A=\frac{1}{1}-\frac{1}{\left(x+1\right)}\)
\(A=\frac{x}{2.\left(x+1\right)}=\frac{8}{17}=\frac{16}{2.17}\)
X=16
X+(1/1.3+1/3.5+1/5.7+...+1/99.101)=100
X+(2/1.3+2/3.5+2/5.7+...+2/99.101)=100
X+(1 -1/3+1/3-1/5+1/5-1/7+...+1/99-1/101)=100
X+(1-1/101)=100
X+100/101=100
X=100-100/101
X=10000/101
b) Vì \(\left|x+\dfrac{1}{1.3}\right| \ge0;\left|x+\dfrac{1}{3.5}\right|\ge0;...;\left|x+\dfrac{1}{97.99}\right|\ge0\)
\(\Rightarrow50x\ge0\Rightarrow x\ge0\)
Khi đó: \(\left|x+\dfrac{1}{1.3}\right|=x+\dfrac{1}{1.3};\left|x+\dfrac{1}{3.5}\right|=x+\dfrac{1}{3.5};...;\left|x+\dfrac{1}{97.99}\right|=x+\dfrac{1}{97.99}\left(1\right)\)
Thay (1) vào đề bài:
\(x+\dfrac{1}{1.3}+x+\dfrac{1}{3.5}+...+x+\dfrac{1}{97.99}=50x\)
\(\Rightarrow\left(x+x+...+x\right)+\left(\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{97.99}\right)=50x\)
\(\Rightarrow49x+\left[\dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{97}-\dfrac{1}{99}\right)\right]=50x\)
\(\Rightarrow49x+\dfrac{16}{99}=50x\)
\(\Rightarrow x=\dfrac{16}{99}\)
Vậy \(x=\dfrac{16}{99}.\)
Bài này khá ez thôi:
a) bạn sửa lại đề rồi làm theo cách làm của b,c,d nhé
b) Ta có: \(\left|x+1,1\right|+\left|x+1,2\right|+\left|x+1,3\right|+\left|x+1,4\right|\ge0\left(\forall x\right)\)
\(\Rightarrow5x\ge0\Rightarrow x\ge0\) khi đó:
\(PT\Leftrightarrow x+1,1+x+1,2+x+1,3+x+1,4=5x\)
\(\Leftrightarrow x=5\)
c,d tương tự nhé
c,\(\left|x+\frac{1}{1.3}\right|+\left|x+\frac{1}{3.5}+\right|+...+\left|x+\frac{1}{97.99}\right|\ge0\forall x\)
\(\Rightarrow50x\ge0\Rightarrow x\ge0\)Khi đó:
\(x+\frac{1}{1.3}+x+\frac{1}{3.5}+...+x+\frac{1}{97.99}=50x\)
\(\Rightarrow49x+\left(\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{97.99}\right)=50x\)
\(\Leftrightarrow x=\frac{1}{2}\left(1-\frac{1}{99}\right)=\frac{49}{99}\)