Question

Aratati ca numarul 2+4+6+...+4028+4030-2015 este un patrat perfect.

Answer 1

2+4+6+...+4028+4030-2015
2+4+6+...+4028+4030=2×(1+2+3+4+.....+2015)=2×[(2015×2016):2]=2015×2016
(1+2+3+4+.....+2015)- suma Gauss care are formula de calcul S=[n(n+1)]:2
S=[2015×(2015+1)]:2=(2015×2016):2=
revenim la
(2+4+6+...+4028+4030)-2015=
=2015×2016-2015=2015×(2016-1)=2015×2015=2015² deci este patrat perfect