Question

Cho ba số a,b,c thõa mãn abc=1. Tính tổng

B= 2016/1+a+ab + 2016/1+b+bc + 2016/1+c+ca

Answer 1

\(B=\dfrac{2016}{1+a+ab}+\dfrac{2016}{1+b+bc}+\dfrac{2016}{1+c+ca}\)

\(=2016\left(\dfrac{1}{1+a+ab}+\dfrac{1}{1+b+bc}+\dfrac{1}{1+c+ca}\right)\)

\(=2016\left(\dfrac{c}{c+ac+abc}+\dfrac{ac}{ac+abc+abc^2}+\dfrac{1}{1+c+ca}\right)\)

Thay \(abc=1\) vào \(B\) ta được :

\(B=2016\left(\dfrac{c}{c+ac+1}+\dfrac{ac}{ac+1+c}+\dfrac{1}{1+c+ca}\right)\)

\(=2016\cdot\dfrac{c+ac+1}{c+ac+1}=2016\)

Vậy : Với \(abc=1\) thì \(B=2016\)