Question

fie x= 5/1+9/2+...+8045/2011-(1+1/2+...+1/2011). Calculati (x/4-2010)^2011

Answer 1

$x = \dfrac{5}{1}+\dfrac{9}{2}+\dfrac{13}{3}+...+\dfrac{8045}{2011}-\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2011}\right) \\ \\=\dfrac{5}{1}+\dfrac{9}{2}+\dfrac{13}{3}+...+\dfrac{8045}{2011}-1-\dfrac{1}{2}-\dfrac{1}{3}-...-\dfrac{1}{2011}\\ \\ =\left(\dfrac{5}{1}-1\right)+\left(\dfrac{9}{2}-\dfrac{1}{2}\right)+\left(\dfrac{13}{3}-\dfrac{1}{3}\right)+...+\left(\dfrac{8045}{2011}-\dfrac{1}{2011}\right)\\ \\ =4+\dfrac{9-1}{2}+\dfrac{13-1}{3}+...+\dfrac{8045-1}{2011}\\\\ = 4+\dfrac{8}{2}+\dfrac{12}{3}+...+\dfrac{8044}{2011}$

$= 4+4+4+\underset{\text{de 2011 ori}}{\underbrace{...}}+4\\= 4\cdot 2011\\ \\ \\ \underline{\text{Astfel:}}\\\left(\dfrac{x}{4}-2010\right)^{2011} = \left(\dfrac{4\cdot 2011}{4}-2010\right)^{2011} = (2011-2010)^{2011} =\\ \\ =1^{2011} = \boxed{1}$