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Nếu x=0x=0:
3x2+2x−13x2+2x−1=3.02+2.0−1=−1=3.02+2.0−1=−1
Nếu x=−1x=−1:
3x2+2x−13x2+2x−1=3(−1)2+2(−1)−1=3−2−1=0=3(−1)2+2(−1)−1=3−2−1=0
Nếu x=13x=13:
3x2+2x−13x2+2x−1=3(13)2+2.13−1=13+23−1=0
Nếu \(x=0\):
\(3x^2+2x-1\)\(=3.0^2+2.0-1=-1\)
Nếu \(x=-1\):
\(3x^2+2x-1\)\(=3\left(-1\right)^2+2\left(-1\right)-1=3-2-1=0\)
Nếu \(x=\frac{1}{3}\):
\(3x^2+2x-1\)\(=3\left(\frac{1}{3}\right)^2+2.\frac{1}{3}-1=\frac{1}{3}+\frac{2}{3}-1=0\)
Cho \(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
so sánh B với \(\frac{3}{4}\)
Ta có:\(\frac{1}{2^2}=\frac{1}{4}\)
\(\frac{1}{3^2}< \frac{1}{2.3}\)
\(\frac{1}{4^2}< \frac{1}{3.4}\)
....
\(\frac{1}{100^2}< \frac{1}{99.100}\)
\(\Leftrightarrow B< \frac{1}{4}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}=\frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(=\frac{1}{4}+\frac{1}{2}-\frac{1}{100}\)
B < \(\frac{1}{4}\) < \(\frac{3}{4}\)
\(\Leftrightarrow B< \frac{3}{4}\)
Ta có: \(\frac{1}{2^2}< \frac{1}{1.2}\)
\(\frac{1}{3^2}< \frac{1}{2.3}\)
....................
.....................
\(\frac{1}{100^2}< \frac{1}{99.100}\)
Nên \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+......+\frac{1}{100^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+.....+\frac{1}{99.100}\)
=> \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+......+\frac{1}{100^2}^2< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+......+\frac{1}{99}-\frac{1}{100}\)
=> \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+......+\frac{1}{100^2}^2^2< 1-\frac{1}{100}=\frac{99}{100}\)
\(\frac{99}{100}\)> \(\frac{3}{4}\)thì sao mà so sánh được
Theo bài ta có:
\(=\frac{\left(1+\frac{2}{3}+\frac{3}{3^2}+...+\frac{99}{3^{98}}+\frac{100}{3^{99}}\right)-\left(\frac{1}{3}+\frac{2}{3^2}+...+\frac{99}{3^{99}}+\frac{100}{3^{100}}\right)}{2}\)
\(=\frac{\left(1-\frac{100}{3^{100}}\right)+\left(\frac{2}{3}-\frac{1}{3}\right)+...+\left(\frac{99}{3^{98}}-\frac{98}{3^{98}}\right)+\left(\frac{100}{3^{99}}-\frac{99}{3^{99}}\right)}{2}\)
\(=\frac{\left(1-\frac{100}{3^{100}}\right)+\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{99}}\right)}{2}< \frac{1+\frac{1}{2}}{2}=\frac{3}{2}:2=\frac{3}{4}\)
Đpcm
Cho \(A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{100}};B=\frac{1}{2}\).so sánh A và B
Lời giải:
$A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{100}}$
$3A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{99}}$
$\Rightarrow 3A-A=1-\frac{1}{3^{100}}$
$\Rightarrow 2A=1-\frac{1}{3^{100}}<1$
$\Rightarrow A< \frac{1}{2}$
$\Rightarrow A< B$
Ta có: \(D=\frac13+\frac{1}{3^2}+\cdots+\frac{1}{3^{100}}\)
=>\(3D=1+\frac13+\frac{1}{3^2}+\cdots+\frac{1}{3^{99}}\)
=>\(3D-D=1+\frac13+\cdots+\frac{1}{3^{99}}-\frac13-\frac{1}{3^2}-\cdots-\frac{1}{3^{100}}\)
=>\(2D=1-\frac{1}{3^{100}}\)
=>\(D=\frac12-\frac{1}{2\cdot3^{100}}\)
\(\frac14=\frac12-\frac14=\frac12-\frac{1}{2\cdot2}\)
Ta có: \(2\cdot3^{100}>2\cdot2\)
=>\(\frac{1}{2\cdot3^{100}}<\frac{1}{2\cdot2}\)
=>\(-\frac{1}{2\cdot3^{100}}>-\frac{1}{2\cdot2}\)
=>\(-\frac{1}{2\cdot3^{100}}+\frac12>-\frac14+\frac12\)
=>\(D>\frac14\)
To quá