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\(\frac{x+1}{99}+\frac{x+2}{99}+\frac{x+3}{99}+\frac{x+4}{99}=-4\)
=>\(\frac{\left(x+1\right)+\left(x+2\right)+\left(x+3\right)+\left(x+4\right)}{99}=-4\)
=> (x+1)+(x+2)+(x+3)+(x+4)=-4.99=-396
=>4x+10=-396
4x=-406
x=-406:4=-101,5
\(\frac{x-1}{99}+\frac{x-2}{98}+\frac{x-5}{95}=3+\frac{1}{99}+\frac{1}{98}+\frac{1}{95}\)
\(\Leftrightarrow\frac{x-1}{99}+\frac{x-2}{98}+\frac{x-5}{95}=1+\frac{1}{99}+1+\frac{1}{98}+1+\frac{1}{95}\)
\(\Leftrightarrow\frac{x-1}{99}+\frac{x-2}{98}+\frac{x-5}{95}=\frac{100}{99}+\frac{99}{98}+\frac{96}{95}\)
\(\Leftrightarrow\left(\frac{x-1}{99}-\frac{100}{99}\right)+\left(\frac{x-2}{98}-\frac{99}{98}\right)+\left(\frac{x-5}{95}-\frac{96}{95}\right)=0\)
\(\Leftrightarrow\frac{x-101}{99}+\frac{x-101}{98}+\frac{x-101}{95}=0\)
\(\Leftrightarrow\left(x-101\right).\left(\frac{1}{99}+\frac{1}{98}+\frac{1}{95}\right)=0\)
\(\Leftrightarrow x-101=0\)
\(\Leftrightarrow x=101\)
\(\frac{x-1}{99}+\frac{x-2}{98}+\frac{x-5}{95}=3+\frac{1}{99}+\frac{1}{98}+\frac{1}{95}\)
\(\Leftrightarrow\frac{x-1}{99}+\frac{x-2}{98}+\frac{x-5}{95}=1+\frac{1}{99}+1+\frac{1}{98}+1+\frac{1}{95}\)
\(\Leftrightarrow\frac{x-1}{99}+\frac{x-2}{98}+\frac{x-5}{95}=\frac{100}{99}+\frac{99}{98}+\frac{96}{95}\)
\(\Leftrightarrow\frac{x-1}{99}+\frac{x-2}{98}+\frac{x-5}{95}-\frac{100}{99}-\frac{99}{98}-\frac{96}{95}=0\)
\(\Leftrightarrow\left(\frac{x-1}{99}-\frac{100}{99}\right)+\left(\frac{x-2}{98}-\frac{99}{98}\right)+\left(\frac{x-5}{95}-\frac{96}{95}\right)=0\)
\(\Leftrightarrow\frac{x-101}{99}+\frac{x-101}{98}+\frac{x-101}{95}=0\)
\(\Leftrightarrow\left(x-101\right)\left(\frac{1}{99}+\frac{1}{98}+\frac{1}{95}\right)=0\)
Do \(\frac{1}{99}+\frac{1}{98}+\frac{1}{95}\ne0\)
Mà \(x-101=0\Leftrightarrow x=101\)
Vậy x = 101
*\(\frac{x}{200}\)=\(\frac{1^2}{1.2}\).\(\frac{2^2}{2.3}\)....\(\frac{99^2}{99.100}\)
=>\(\frac{x}{200}\)=\(\frac{1}{2}\).\(\frac{2}{3}\)....\(\frac{99}{100}\)
=>\(\frac{x}{200}\)=\(\frac{1}{100}\)
=>100x=200
=>x=2
\(\left(x+\frac{1}{2}\right)+\left(x+\frac{1}{2}\right)+\left(x+\frac{1}{2}\right)=1\)
\(\Leftrightarrow3x+\left(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\right)=1\)
\(\Leftrightarrow3x+\frac{3}{2}=1\)
\(\Leftrightarrow3x=-\frac{1}{2}\)
\(\Leftrightarrow x=-\frac{1}{2}\div3=-\frac{1}{6}\)
Sửa đề \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{x.\left(x+1\right)}=\frac{99}{100}\)
\(\Leftrightarrow1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2}-\frac{1}{x+1}=\frac{99}{100}\)
\(\Leftrightarrow1-\frac{1}{x+1}=\frac{99}{100}\)
\(\Leftrightarrow\frac{1}{x+1}=\frac{1}{100}\)
\(\Leftrightarrow x=99\)
a) => ( x + 1/2 ) . 3 = 1
=> 3x + 3/2 = 1
=> 3x = 1 - 3/2
=> 3x = -1/2
=> x = -1/2 : 3 = -1/6
Bài 3:
a,Đặt A = \(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\)
A = \(\frac{1}{2}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+\frac{1}{2^5}-\frac{1}{2^6}\)
2A = \(1-\frac{1}{2}+\frac{1}{2^2}-\frac{1}{2^3}+\frac{1}{2^4}-\frac{1}{2^5}\)
2A + A = \(\left(1-\frac{1}{2}+\frac{1}{2^2}-\frac{1}{2^3}+\frac{1}{2^4}-\frac{1}{2^5}\right)+\left(\frac{1}{2}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+\frac{1}{2^5}-\frac{1}{2^6}\right)\)
3A = \(1-\frac{1}{2^6}\)
=> 3A < 1
=> A < \(\frac{1}{3}\)(đpcm)
b, Đặt A = \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+...+\frac{99}{3^{99}}-\frac{100}{3^{100}}\)
3A = \(1-\frac{2}{3}+\frac{3}{3^2}-\frac{4}{4^3}+...+\frac{99}{3^{98}}-\frac{100}{3^{99}}\)
3A + A = \(\left(1-\frac{2}{3}+\frac{3}{3^2}-\frac{4}{4^3}+...+\frac{99}{3^{98}}-\frac{100}{3^{99}}\right)-\left(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+...+\frac{99}{3^{99}}-\frac{100}{3^{100}}\right)\)
4A = \(1-\frac{1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+...+\frac{1}{3^{98}}-\frac{1}{3^{99}}-\frac{100}{3^{100}}\)
=> 4A < \(1-\frac{1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+...+\frac{1}{3^{98}}-\frac{1}{3^{99}}\) (1)
Đặt B = \(1-\frac{1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+...+\frac{1}{3^{98}}-\frac{1}{3^{99}}\)
3B = \(3-1+\frac{1}{3}-\frac{1}{3^2}+...+\frac{1}{3^{97}}-\frac{1}{3^{98}}\)
3B + B = \(\left(3-1+\frac{1}{3}-\frac{1}{3^2}+...+\frac{1}{3^{97}}-\frac{1}{3^{98}}\right)+\left(1-\frac{1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+...+\frac{1}{3^{98}}-\frac{1}{3^{99}}\right)\)
4B = \(3-\frac{1}{3^{99}}\)
=> 4B < 3
=> B < \(\frac{3}{4}\) (2)
Từ (1) và (2) suy ra 4A < B < \(\frac{3}{4}\)=> A < \(\frac{3}{16}\)(đpcm)
\(\frac{1}{x}\)+\(\frac{2}{x}\)+\(\frac{3}{x}\)+....+\(\frac{99}{x}\)+\(\frac{100}{x}\)=5050
\(\frac{1+2+3+....+99+100}{x}\)=5050
\(\frac{5050}{x}\)=5050
\(\frac{5050}{x}\)=\(\frac{5050}{1}\)
\(\Rightarrow x=1\)
Vậy x = 1