Question

       Tính

cho f(x) = x^2016 - 2013x^2015+ 2013x^2014 -2013x^2013 + ........+ 2013x^2 -2013x +2013

với f (2012)

Answer 1

Đặt \(g\left(x\right)=x^{2015}-x^{2014}+x^{2013}-...+x-1\)

Dễ thấy: \(f\left(x\right)=x^{2016}-2013\times g\left(x\right)\Rightarrow f\left(2012\right)=2012^{2016}-2013\times g\left(2012\right)\)(a)

Ta có: \(\left(x+1\right)\times g\left(x\right)=\left(x+1\right)\left(x^{2015}-x^{2014}+x^{2013}-...+x-1\right)\)

\(\Rightarrow\left(x+1\right)\times g\left(x\right)=x^{2016}-1\)

\(\Rightarrow\left(2012+1\right)\times g\left(2012\right)=2012^{2016}-1\)hay: \(2013\times g\left(2012\right)=2012^{2016}-1\)

Thay vào (a) ta có: \(f\left(2012\right)=2012^{2016}-\left(2012^{2016}-1\right)=1\).