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\(B=\dfrac{\sqrt{2}-1}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}+\dfrac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}+...+\dfrac{\sqrt{100}-\sqrt{99}}{\left(\sqrt{100}-\sqrt{99}\right)\left(\sqrt{100}+\sqrt{99}\right)}\)
\(=\dfrac{\sqrt{2}-1}{1}+\dfrac{\sqrt{3}-\sqrt{2}}{1}+...+\dfrac{\sqrt{100}-\sqrt{99}}{1}\)
\(=\sqrt{100}-1=9\)
\(x^3+3.9x^2+3.9^2x+9^3=0\)
\(\Leftrightarrow\left(x+9\right)^3=0\)
\(\Leftrightarrow x=-9\)
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B = ( 1/2^2 - 1) . ( 1/3^2 - 1) . ( 1/4^2 - 1) ... ( 1/98^2 - 1) . ( 1/99^2 - 1)
B = -3/2^2 . ( -8/3^2) . ( -15/4^2) .... ( -9603/98^2) . ( -9800/99^2)
Tích B gồm ( 99 - 2) : 1 + 1 = 98 ( số hạng), mỗi số hạng đều mang dấu âm nên khi tính ra kết quả tích B mang dấu dương
=> B = 3/2^2 . 8/3^2 . 15/4^2 ... 9603/98^2 . 9800/99^2
B = 1.3/2.2 . 2.4/3.3 . 3.5/4.4 ... 97.99/98.98 . 98.100/99.99
B = 1.2.3...97.98/2.3.4...98.99 . 3.4.5...99.100/2.3.4...98.99
B = 1/99 . 100/2
B = 1/99 . 50 = 50/99
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`1/(\sqrt100+\sqrt99) + 1/(\sqrt99+\sqrt98)+....+1/(\sqrt2+1)`
`=(\sqrt100-\sqrt99)/(100-99) + (\sqrt99-\sqrt98)/(99-98) + ... + (\sqrt2-1)/(2-1)`
`=\sqrt100 - \sqrt99 + \sqrt99 -\sqrt98 +... + \sqrt2-1`
`=\sqrt100-1`
`=10-1=9`
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\(=\dfrac{\left(1+\dfrac{99}{2}+1+\dfrac{98}{3}+...+1+\dfrac{1}{100}+1\right)}{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{101}}-2\)
\(=\dfrac{\dfrac{101}{2}+\dfrac{101}{3}+...+\dfrac{101}{100}+\dfrac{101}{101}}{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{101}}-2\)
=101-2
=99
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