tim a biet 1/2x3 + 1/3x4 + 1/4x5 +....+1/ax(a+1)=299/600
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\(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{x.\left(x+1\right)}=\frac{299}{600}\)
\(\Rightarrow\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{299}{600}\)
\(\Rightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{299}{600}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{299}{600}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{600}\)
\(\Rightarrow x+1=600\)
\(\Rightarrow x=600-1\)
\(\Rightarrow x=599\)
\(Vậy\) \(x=599\)
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\(A=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{7}-\dfrac{1}{8}=\dfrac{1}{2}-\dfrac{1}{8}=\dfrac{3}{8}\)
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Đặt \(A=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}\)
\(\Leftrightarrow A=\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+...+\left(\frac{1}{99}-\frac{1}{100}\right)\)
\(\Leftrightarrow A=\frac{1}{2}-\frac{1}{100}\)
\(\Leftrightarrow A=\frac{49}{100}\)
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\(\frac{1}{2x3}+\frac{1}{3x4}+\frac{1}{4x5}+...+\frac{1}{98x99}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{98}-\frac{1}{99}\)
\(=\frac{1}{2}-\frac{1}{99}\)
\(=\frac{99}{198}-\frac{2}{198}\)
\(=\frac{97}{198}\)
\(\frac{A}{198}=\frac{97}{198}=>A=198x97:198=97\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{6}-\dfrac{1}{7}=\dfrac{1}{2}-\dfrac{1}{7}=\dfrac{5}{14}\)
\(=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{20}-\dfrac{1}{21}=\dfrac{21-2}{42}=\dfrac{19}{42}\)
Lời giải:
Gọi biểu thức số 1 là A và số 2 là B
\(A=\frac{3-2}{2\times 3}+\frac{4-3}{3\times 4}+\frac{5-4}{4\times 5}+\frac{6-5}{5\times 6}+\frac{7-6}{6\times 7}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}\)
\(=\frac{1}{2}-\frac{1}{7}=\frac{5}{14}\)
B tương tự A:
\(B=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+....+\frac{1}{20}-\frac{1}{21}\)
\(=\frac{1}{2}-\frac{1}{21}=\frac{19}{42}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
1/2x3 +1/3x4 +..........+ 1/ax(a+1)=299/600
=>1/2-1/3+1/3-1/4+.........+ 1/a -1/a+1=299/600
=>1/2-1/a+1=299/600
=>a-1/2a=299/600
=>a=300