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a fost răspuns

A) Aratati ca numarul a=2003+2*(1+2+...2002) este patrat perfect.
B) Aratati ca b=1+3+5+...2011 este patra perfect.
C) Aratati ca c=81+2*81+3*81+...+49*81 este patrat perfect.

Răspuns :

Renatemambouko
a) a=2003+2*(1+2+...2002) =
=2003+2*2002*2003/2=2003+2002*2003=2003*(1+2002)=2003*2003=2003²

b) b=1+3+5+...2011=1+2+3+4+5+....+2010+2011-2-4-....-2010=
=2010*2011/2-2(1+2+...+1005)=
=1005*2011-2*1005*1006/2=
=1005*2011-1005*1006=
=1005*(2011-1006)=
=1005*1005=1005²

 c) c=81+2*81+3*81+...+49*81 =81(1+2+3+...+49)=81*49*50/2=81*49*25=
=9² *7² *5²=(9 *7*5)²=315²





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