Question

[Nguyễn Tuấn Anh]

Bài 1 :

Cho A= 1 - 1/2+ 1/3 - 1/4 + .... - 1/2012

B = 1/1007 + 1/2008 + .... + 1/2012

Tính (A/B)²⁰¹³

Answer 1

[.]

Ta có: \(A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...-\frac{1}{2012}\)

\(=\left(1+\frac{1}{3}+...+\frac{1}{2011}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2012}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2012}\right)-2\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2012}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2012}\right)-\left(1+\frac{1}{2}+...+\frac{1}{1006}\right)\)

\(=\frac{1}{1007}+\frac{1}{1008}+...+\frac{1}{2012}\)

\(\Rightarrow A=B\Rightarrow\frac{A}{B}=1\Rightarrow\left(\frac{A}{B}\right)^{2013}=1\)

Vậy \(\left(\frac{A}{B}\right)^{2013}=1\).