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\(\dfrac{1}{\left(x+2000\right)\left(x+2001\right)}+\dfrac{1}{\left(x+2001\right)\left(x+2002\right)}+...+\dfrac{1}{\left(x+2009\right)\left(x+2010\right)}=\dfrac{10}{11}\\ \Leftrightarrow\dfrac{1}{x+2000}-\dfrac{1}{x+2001}+\dfrac{1}{x+2001}-\dfrac{1}{x+2002}+...+\dfrac{1}{x+2009}-\dfrac{1}{x+2010}=\dfrac{10}{11}\)
\(\Leftrightarrow\dfrac{1}{x+2000}-\dfrac{1}{x+2010}=\dfrac{10}{11}\\ \Leftrightarrow\dfrac{x+2010-x-2000}{\left(x+2000\right)\left(x+2010\right)}=\dfrac{10}{11}\)
\(\Leftrightarrow\dfrac{1}{x+2000}-\dfrac{1}{x+2010}=\dfrac{10}{11}\\ \Leftrightarrow\dfrac{10}{\left(x+2000\right)\left(x+2010\right)}=\dfrac{10}{11}\\ \Leftrightarrow\left(x+2000\right)\left(x+2010\right)=11\\ \Leftrightarrow...\)
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Ta có:
\(A=x^{2002}-x+x^{2000}-x^2+x^2+x+1=x^{2001}-1.x+x^2.x^{1998}-1+x^2+x+1\)
Lại có:
\(x^{2001}-1\)và \(x^{1998}-1⋮x^3-1⋮x^2+x+1\RightarrowĐPCM\)
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Lời giải:
$x^{2002}+x^{2000}+1=(x^{2002}-x)+(x^{2000}-x^2)+(x^2+x+1)$
$=x(x^{2001}-1)+x^2(x^{1998}-1)+(x^2+x+1)$
$=x[(x^3)^{667}-1]+x^2[(x^3)^{666}-1]+(x^2+x+1)$
$=x(x^3-1)[(x^3)^{666}+...+x^3+1]+x^2(x^3-1)[(x^3)^{665}+...+x^3+1]+(x^2+x+1)$
$=x(x-1)(x^2+x+1)[(x^3)^{666}+...+x^3+1]+x^2(x-1)(x^2+x+1)[(x^3)^{665}+...+x^3+1]+(x^2+x+1)$
$=(x^2+x+1)[x(x-1)[(x^3)^{666}+...+x^3+1]+x^2(x-1)[(x^3)^{665}+...+x^3+1]+1]\vdots x^2+x+1$
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\(\frac{1}{x+2000}-\frac{1}{x+2007}=\frac{7}{8}\)
\(\frac{8\left(x+2007\right)}{8\left(x+2000\right)\left(x+2007\right)}-\frac{8\left(x+2000\right)}{8\left(x+2000\right)\left(x+2007\right)}=\frac{7\left(x+2000\right)\left(x+2007\right)}{8\left(x+2000\right)\left(x+2007\right)}\)
\(8x+8.2007-8x+8.2000=7\left(x^2+4007x+2000.2007\right)\)
\(8.7-7\left(x^2+4007x+2000.2007\right)=0\)
\(7\left(8-x^2-4007x-2000.2007\right)=0\)
\(8-x^2-4007x-2000.2007=0\)
\(x^2+4007x+4013992=0\)
\(\left(x^2+2008x\right)+\left(1999x+4013992\right)=0\)
\(\left(x+2008\right)\left(x+1999\right)=0\)
\(\hept{\begin{cases}x=-2008\\x=-1999\end{cases}}\)
\(\frac{1}{\left(x+2000\right)\left(x+2001\right)}+\frac{1}{\left(x+2001\right)\left(x+2002\right)}+\frac{1}{\left(x+2006\right)\left(x+2007\right)}=\frac{7}{8}\)
\(\frac{1}{x+2000}-\frac{1}{x+2001}+\frac{1}{x+2001}-\frac{1}{x+2002}+...+\frac{1}{x+2006}-\frac{1}{x+2007}=\frac{7}{8}\)
\(\frac{1}{x+2000}-\frac{1}{x+2007}=\frac{7}{8}\)
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áp dụng : x3m+2+x3n+1+1 luon chia hết cho (x2+x+1) voi71 m,n E N
\(x^{2000}\left(x^2+x+1\right)-\left(x^{2001}-1\right)\)số hạng thứ nhất hiển nhiên chia hết cho A=x^2+x+1 khác 0 với mọi x
xét: \(C=x^{2001}-1\)
Nếu x=1 => C=0 hiển nhiên C chia hết cho A
nếu x khác 1
\(B=\left(1+x+x^2+...+x^{2000}\right)=\frac{\left(x^{2001}-1\right)}{\left(x-1\right)}=\frac{C}{x-1}\)
B có 2001 số hạng chia hết cho 3 => ghép 3 số hạng liên tiếp có
\(B=\left(1+x+x^2\right)+x^3\left(1+x+x^2\right)+x^6\left(1+x+x^2\right)+..+x^{1998}\left(1+x+x^2\right)\)
Hiển nhiên B chia hết cho A
C=B(x-1) chia hết cho A do B chia hết cho A
=> DPCM
oke bạn