Chứng minh 3^2014+3^2013-3^2012
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Ta có :
\(3^{2014}+3^{2013}-3^{2012}\)
\(=3^{2012}\left(3^2+3-1\right)\)
\(=3^{2012}.11\)
\(\Rightarrow3^{2014}+3^{2013}-3^{2012}\)
\(\RightarrowĐPCM\)
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\(3^{2014}-3^{2013}+3^{2012}=3^{2012}\left(9-3+1\right)\)
\(=3^{2012}\cdot7=3^{2010}\cdot63⋮63\)
Dpcm
32014 - 32013 + 32012
= 32012 x 32 - 32012 x 3 + 32012 x 1
= 32012 x 9 - 32012 x 3 + 32012 x 1
= 32012 x (9 - 3 + 1)
= 32012 x 7
= 32010 x 32 x 7
= 32010 x 9 x 7
= 32010 x 63
Mà 63 \(⋮\) 63 nên 32010 x 63 \(⋮\) 63 => 32014 - 32013 + 32012 \(⋮\)63
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\(\frac{\frac{1}{2012}+\frac{1}{2013}-\frac{1}{2014}}{\frac{5}{2012}+\frac{5}{2013}-\frac{5}{2014}}-\frac{\frac{2}{2013}+\frac{2}{2014}-\frac{2}{2015}}{\frac{3}{2013}+\frac{3}{2014}-\frac{3}{2015}}\)
=\(\frac{\frac{1}{2012}+\frac{1}{2013}-\frac{1}{2014}}{5\left(\frac{1}{2012}+\frac{1}{2013}-\frac{1}{2014}\right)}-\frac{2\left(\frac{1}{2013}+\frac{1}{2014}-\frac{1}{2015}\right)}{3\left(\frac{1}{2013}+\frac{1}{2014}-\frac{1}{2015}\right)}=\frac{1}{5}-\frac{2}{3}=\frac{3}{15}-\frac{10}{15}=-\frac{7}{15}\)
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Đề \(\Rightarrow a^{2014}+b^{2014}-2\left(a^{2013}+b^{2013}\right)+a^{2012}+b^{2012}=0\)
\(\Leftrightarrow a^{2012}\left(a^2-2a+1\right)+b^{2012}\left(b^2-2b+1\right)=0\)
\(\Leftrightarrow a^{2012}\left(a-1\right)^2+b^{2012}\left(b-1\right)^2=0\)
\(\Leftrightarrow\left(a=0\text{ hoặc }a=1\right)\text{ và }\left(b=0\text{ hoặc }b=1\right)\)
\(+a=0\text{ hoặc }a=1\text{ thì }a^{2014}=a^{2010}\)
\(+b=0\text{ hoặc }b=1\text{ thì }b^{2014}=b^{2010}\)
Suy ra \(a^{2014}+b^{2014}=a^{2010}+b^{2010}\)
Đề bị thiếu rồi bạn ơi? nhi đặng