Question

5.Aratati ca numarul 2005+2*(1+2+3+...+2004) este patrat perfect
Varog ajutatima..

Answer 1

suma lui Gauss zice ca 1+2+3+4+....+n=n*(n+1):2.

in cazul nostrum n=2004, deci 1+2+3+....+2004=2004*(2004+1):2=2004*2005:2,

de unde 2005+2*(1+2+3+....+2004)=2005+2*(2004*2005):2=2005+2004*2005=2005+(1+2004)=2005+2005 este patrat perfect

Answer 2

2005+2X(1+2+3...2004)=2005+2x2004(2004+1)/2=2005+2004x2005=2005(2004+1)=2005^2    este patrat perfect