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\(B=1-5+5^{^2}-5^{^3}+...-5^{^{99}}+5^{^{100}}\)
\(5B=5-5^{^2}+5^{^3}-5^{^4}+...-5^{^{100}}+5^{^{101}}\)
\(5B+B=\left(5-5^{^2}+5^{^3}-5^{^4}+...-5^{^{100}}+5^{^{101}}\right)+\left(1-5+5^{^2}-5^{^3}+...-5^{^{99}}+5^{^{100}}\right)\)
\(6B=5^{^{101}}+1\)
\(B=\dfrac{5^{^{101}}+1}{6}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(a,\left|x+2\right|=0\)
\(\Leftrightarrow x+2=0\)
\(\Leftrightarrow x=-2\)
\(b,\left|x-5\right|=\left|-7\right|\)
\(\Leftrightarrow\left|x-5\right|=7\)
\(\Leftrightarrow\left[{}\begin{matrix}x-5=7\\x-5=-7\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=12\\x=-2\end{matrix}\right.\)
\(c,\left(7-x\right)-\left(25+7\right)=-25\)
\(\Leftrightarrow7-x-32=-25\)
\(\Leftrightarrow x=0\)
\(d,\left|x-3\right|=\left|5\right|+\left|-7\right|\)
\(\Leftrightarrow\left|x-3\right|=12\)
\(\Leftrightarrow\left[{}\begin{matrix}x-3=12\\x-3=-12\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=15\\x=-9\end{matrix}\right.\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^8}\)
\(3A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^7}\)
\(3A-A=\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^7}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^8}\right)\)
\(2A=1-\frac{1}{3^8}\)
\(A=\frac{1}{2}-\frac{1}{2.3^8}\)
\(F=2+2^2+2^3+...+2^{100}\)
\(2F=2^2+2^3+2^4+...+2^{101}\)
\(2F-F=\left(2^2+2^3+...+2^{101}\right)-\left(2+2^2+2^3+...+2^{100}\right)\)
\(F=2^{101}-2\)
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\(E=3^0+3^1+3^2+...+3^{100}\)
\(E=1+3+3^2+...+3^{100}\)
\(3E=3+3^2+...+3^{101}\)
\(3E-3E=\left(3+3^2+...+3^{101}\right)-\left(1+3+3^2+...+3^{100}\right)\)
\(2E=3^{101}-1\)
\(E=\frac{3^{101}-1}{2}\)
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