Question

[Hồ Quế Ngân]

1) Tính A = \(\dfrac{x^{98}+x^{97}+....+x+1}{x^{32}+x^{31}+.,..+x+1}\) tại x = 2

2) Rút gọn: B = \(\dfrac{1}{1+\sqrt{5}}+\dfrac{1}{\sqrt{2}+\sqrt{6}}+....+\dfrac{1}{\sqrt{2009}+\sqrt{2013}}+\dfrac{1}{\sqrt{2010}+\sqrt{2014}}\)

3) Cho x,y thỏa \(x^{671}+y^{671}=0,67\) ; \(x^{1342}+y^{1342}=1,34\) Tính A=\(x^{2013}+y^{2013}\)

Answer 1

[Phương An]

Đặt \(\left\{{}\begin{matrix}x^{671}=a\\y^{671}=b\end{matrix}\right.\). Bài toán trở thành

Cho \(a+b=0,67\) và \(a^2+b^2=1,34\). Tính \(A=a^3+b^3\)

Giải:

\(a^2+2ab+b^2=0,4489\)

\(\Rightarrow ab=\dfrac{0,4489-1,34}{2}=-0,44555\)

\(A=a^3+b^3=\left(a+b\right)^3-3ab\left(a+b\right)=1,1963185\)

Answer 2

[Phương An]

\(4B=\dfrac{4}{\sqrt{5}+1}+\dfrac{4}{\sqrt{6}+\sqrt{2}}+...+\dfrac{4}{\sqrt{2014}+\sqrt{2010}}\)

\(=\dfrac{4\left(\sqrt{5}-1\right)}{5-1}+\dfrac{4\left(\sqrt{6}-\sqrt{2}\right)}{6-2}+...+\dfrac{4\left(\sqrt{2014}-\sqrt{2010}\right)}{2014-2010}\)

\(=\sqrt{5}-1+\sqrt{6}-\sqrt{2}+...+\sqrt{2014}-\sqrt{2010}\)

\(=-1-\sqrt{2}-\sqrt{3}-\sqrt{4}+\sqrt{2011}+\sqrt{2012}+\sqrt{2013}+\sqrt{2014}\)

\(\Rightarrow B=...\)