Question

Aflați x din egalitățile: a)2+4+6+8+...+2010+2012+x=1+3+5+7+...+2011+2013.

Answer 1

2+4+6+8+...+2012=2*(1+2+3+4...+1006)=2*1006*1007/2=1006*1007, iar
1+3+5+7+...+2013=1+(1+2*1)+(1+2*2)+(1+2*3)+...+(1+2*1006)=1+1006+2*(1+2+3+...+1006)=1007+2*1006*1007/2=1007+1006*1007=1007*(1+1006)=1007^2.

Deci ecuatia din enunt este echivalenta cu 1006*1007+x=1007^2 <=> x=1007^2-1006*1007 <=> x=1007*(1007-1006) <=> x=1007.