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\(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+...+\frac{1}{x\cdot\left(x+1\right)}=2\)
\(\Leftrightarrow\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}+...+\frac{1}{x\cdot\left(x+1\right)}=2\)
\(\Leftrightarrow\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{x}-\frac{1}{x+1}=2\)
\(\Leftrightarrow\frac{1}{2}-\frac{1}{x+1}=2\)
\(\Leftrightarrow\frac{1}{x+1}=\frac{1}{2}-2\)
\(\Leftrightarrow\frac{1}{x+1}=\frac{-3}{2}\)
\(\Leftrightarrow\frac{-3}{-3x-3}=\frac{-3}{2}\)
\(\Leftrightarrow-3x-3=2\)
\(\Leftrightarrow-3x=2+3\)
\(\Leftrightarrow-3x=5\)
\(\Leftrightarrow x=\frac{-5}{3}\)
Vậy \(x=\frac{-5}{3}\)
Ta có:\(2^{30}=\left(2^3\right)^{10}=8^{10}< 9^{10}=\left(3^2\right)^{10}=3^{20}\)
\(3^{30}=3^{20}.3^{10}< 3^{20}.4^{10}=3^{20}.\left(2^2\right)^{10}=3^{20}.2^{20}=\left(3.2\right)^{20}=6^{20}\)
\(4^{30}=4^{20}.4^{10}=4^{20}.\left(2^2\right)^{10}=4^{20}.2^{20}=\left(4.2\right)^{20}=8^{20}\)
nên \(2^{30}+3^{30}+4^{30}< 3^{20}+6^{20}+8^{20}\)
ta có : \(\left(4x-1\right)^{30}=\left(4x-1\right)^{20}\)(=)\(\left(4x-1\right)^{30}-\left(4x-1\right)^{20}=0\)(=)\(\left(4x-1\right)^{20}\left[\left(4x-1\right)^{10}-1\right]=0\)(=)\(\orbr{\begin{cases}\left(4x-1\right)^{20}=0\\\left[\left(4x-1\right)^{10}-1\right]=0\end{cases}}\)(=)\(\orbr{\begin{cases}4x-1=0\\\left(4x-1\right)^{10}=1\end{cases}}\)(=)\(\orbr{\begin{cases}4x=1\\\begin{cases}4x-1=1\\4x-1=-1\end{cases}\end{cases}}\)(=)\(\orbr{\begin{cases}x=\frac{1}{3}\\\begin{cases}4x=2\\4x=0\end{cases}\end{cases}}\)\(\orbr{\begin{cases}x=\frac{1}{4}\\\begin{cases}x=\frac{1}{2}\\x=0\end{cases}\end{cases}}\)
\(\left(4x-1\right)^{30}=\left(4x-1\right)^{20}\)
\(4x^{30}-1^{30}=4x^{20}-1^{20}\)
\(4x^{30}-4x^{20}=-1+1\)
\(4x^{20}\left(x^{10}-1\right)=0\)
\(\Rightarrow\orbr{\begin{cases}4x^{20}=0\\x^{10}-1=0\end{cases}\Rightarrow\orbr{\begin{cases}x^{20}=0\\x^{10}=1\end{cases}\Rightarrow}\orbr{\begin{cases}x=0\\x=\pm1\end{cases}}}\)
hok tốt!!
Ta có \(\left(4x-1\right)^{30}=\left(4x-1\right)^{20}\)
<=> \(\left(4x-1\right)^{30}-\left(4x-1\right)^{20}=0\)
<=> \(\left(4x-1\right)^{20}.\left[\left(4x-1\right)^{10}-1\right]=0\)
<=> \(\orbr{\begin{cases}\left(4x-1\right)^{20}=0\\\left(4x-1\right)^{10}-1=0\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}4x-1=0\\\left(4x-1\right)^{10}=1\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}4x=1\\4x-1=1;4x-1=-1\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{1}{4}\\4x=2;4x=0\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{1}{4}\\x=2;x=0\end{cases}}\)
Vậy \(x\in\left\{0;2;\frac{1}{4}\right\}\)
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Hc tốt
\(\left(4x-1\right)^{30}=\left(4x-1\right)^{20}\)
\(\Leftrightarrow\left(4x-1\right)^{30}-\left(4x-1\right)^{20}=0\)
\(\Leftrightarrow\left(4x-1\right)^{20}\left[\left(4x-1\right)^{10}-1\right]=0\)
\(\Leftrightarrow\orbr{\begin{cases}\left(4x-1\right)^{20}=0\\\left(4x-1\right)^{10}-1=0\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}4x-1=0\\\left(4x-1\right)^{10}=1\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}4x=1\\4x-1=-1;1\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=\frac{1}{4}\\x=0\\x=\frac{1}{2}\end{cases}}\)
Vậy \(x=\frac{1}{4};0;\frac{1}{2}\)
P/s : phần \(\Leftrightarrow\hept{\begin{cases}x=\frac{1}{4}\\x=0\\x=\frac{1}{2}\end{cases}}\) thay dấu \(\hept{\begin{cases}\\\\\end{cases}}\) thành dấu \(\orbr{\begin{cases}\\\end{cases}}\) nhé!
\(\Leftrightarrow\orbr{\begin{cases}4x=1\\4x-1=\pm1\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=\frac{1}{4}\\\orbr{\begin{cases}x=0\\x=\frac{1}{2}\end{cases}}\end{cases}}\)
\(\left(4x-1\right)^{30}=\left(4x-1\right)^{20}\)
\(\Leftrightarrow\left(4x-1\right)^{30}-\left(4x-1\right)^{20}=0\)
\(\Leftrightarrow\left(4x-1\right)^{20}.\left[\left(4x-1\right)^{10}-1\right]=0\)
\(\Leftrightarrow\orbr{\begin{cases}\left(4x-1\right)^{20}=0\\\left(4x-1\right)^{10}-1=0\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}4x-1=0\\\left(4x-1\right)^{10}=1\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{1}{4}\\4x-1=\pm1\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{1}{2}\\x=0\end{cases}}\)
Vậy x = 1/4 hoặc 1/2 hoặc 0
\(\Leftrightarrow\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{x}-\dfrac{1}{x+1}=\dfrac{199}{600}\)
\(\Leftrightarrow\dfrac{1}{x+1}=\dfrac{1}{600}\)
=>x+1=600
hay x=599
\(\Leftrightarrow-2x+\dfrac{3}{20}=1-\dfrac{1}{2}+1-\dfrac{1}{6}+...+1-\dfrac{1}{110}\)
\(\Leftrightarrow-2x+\dfrac{3}{20}=10-\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{10}-\dfrac{1}{11}\right)\)
\(\Leftrightarrow-2x+\dfrac{3}{20}=10-\dfrac{10}{11}=\dfrac{100}{11}\)
=>-2x=1967/220
hay x=-1967/440
I 1-x I = 30-20 =10
1-x = \(\pm\)10
TH1:1-x=-10
x=11
TH2: 1-x=10
x=-9