cho a^2=b^2+c^2,So sánh a va b+c, a^3 và b^3 +c^3?
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a^2 = b^2 + c^2 (1)
=> a^2 = (b+c)^2 - 2bc
=> a^2 <= (b+c)^2
=> a <= b+c (2)
Nhân (1) với (2), vế theo vế ta có:
a^3 = b^3 + c^3 + bc(b+c)
=> a^3 >= b^3 + c^3
\(\left(a+b+c\right)^2=a^2+b^2+c^2\)
\(\Rightarrow a^2+b^2+c^2+2ab+2bc+2ac=a^2+b^2+c^2\)
\(\Rightarrow2\left(ab+bc+ac\right)=0\)
\(\Rightarrow ab+bc+ac=0\)
\(\Rightarrow\frac{\left(a+b+c\right)}{abc}=0\)
\(\Rightarrow\frac{ab}{abc}+\frac{bc}{abc}+\frac{ac}{abc}=0\)
\(\Rightarrow\frac{1}{c}+\frac{1}{a}+\frac{1}{b}=0\)
\(\Rightarrow\frac{1}{a}+\frac{1}{b}=\frac{-1}{c}\)
\(\Rightarrow\left(\frac{1}{a}+\frac{1}{b}\right)^3=\left(\frac{-1}{c}\right)^3\)
\(\Rightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{3}{ab}\left(\frac{1}{a}+\frac{1}{b}\right)=-\frac{1}{c^3}\)
\(\Rightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{3}{ab}.\left(-\frac{1}{c}\right)=0\)
\(\Rightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}-\frac{3}{ab}=0\)
\(\Rightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=\frac{3}{abc}\left(đpcm\right)\)
\(\left(a+b+c\right)^2=a^2+b^2+c^2\Rightarrow a^2+b^2+c^2+2ab+2bc+2ac=a^2+b^2+c^2\Rightarrow ab+bc+ac=0\)
\(\Rightarrow\frac{ab+bc+ac}{abc}=0\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Rightarrow\left(\frac{1}{a}\right)^3+\left(\frac{1}{b}\right)^3+\left(\frac{1}{c}\right)^3=3.\frac{1}{a}.\frac{1}{b}.\frac{1}{c}\)
\(\Rightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=\frac{3}{abc}\)
a) Xin lỗi bạn nhé !!!
b) 2010^2 và 2009.2011
<=> (2009+1).2010 và 2009.(2010+1)
<=> 2009.2010+2010 > 2009.2010+2009
=> 2010^2 > 2009 . 2011
c)
\(3^{450}=3^{3\cdot150}=\left(3^3\right)^{150}=27^{150}\)
\(5^{300}=5^{2\cdot150}=\left(5^2\right)^{150}=25^{150}\)
Vì \(27^{150}>25^{150}\)
Nên \(3^{450}>5^{300}\)
a) A = 2 + 22 + ... + 22010
= ( 2 + 22 ) + ( 23 + 24 ) + ... + ( 22009 + 22010 )
= 2.(1+2) + 23.(1+2) + ... + 22009.(1+2)
= 2.3 + 23.3 + ... + 22009.3 chia hết cho 3
A = 2 + 22 + ... + 22010
= ( 2 + 22 + 23 ) + ( 24 + 25 + 26 ) + ... + ( 22008 + 22009 + 22010 )
= 2.(1+2+22) + 24.(1+2+22) + ... + 22008.(1+2+22)
= 2.7 + 24.7 + ... + 22008.7 chia hết cho 7
b) Xét A = 2009.2011
= (2010-1) . (2010+1)
= 2010.2010 + 1.2010 - 1.2010 - 1.1
= 2010.2010 - 1
B = A - 1
Vậy B < A
c) Ta có : 3450 = 35.90 = 1590
5300 = 53.100 = 15100
Vì 1590 < 15100 nên 3450 < 5300 hay A < B