Question

cho a,b dương và \(a^{2000}+b^{2000}=a^{2001}+b^{2001}=a^{2002}+b^{2002}\)tính \(a^{2011}+b^{2011}\)

Answer 1

\(a^{2000}+b^{2000}=a^{2001}+b^{2001}\)

\(\Leftrightarrow a^{2000}\left(a-1\right)+b^{2000}\left(b-1\right)=0\left(1\right)\)

\(a^{2001}+b^{2001}=b^{2002}+a^{2002}\)

\(\Leftrightarrow a^{2001}\left(a-1\right)+b^{2001}\left(b-1\right)=0\left(2\right)\)

Trừ vế theo vế ta được:

\(\left(a-1\right)\left(a^{2001}-a^{2000}\right)+\left(b-1\right)\left(b^{2001}-b^{2000}\right)=0\)

\(\Leftrightarrow\left(a-1\right)a^{2000}\left(a-1\right)+\left(b-1\right)b^{2000}\left(b-1\right)=0\)

\(\Leftrightarrow\left(a-1\right)^2a^{2000}+\left(b-1\right)^2b^{2000}=0\)

Mà a,b dương\(\Rightarrow a=b=1\)

\(\Rightarrow a^{2011}+b^{2011}=2\)