Question

a)Aratati ca numarul a=2003+2x(1+2+...+2002) este patrat perfect
b)Aratati ca numarul b=1+3+5+...+2011 este patrat perfect
c)Aratati ca numarul a=81+2x81+3x81+...49x81 este patrat perfect

Answer 1

a)

2003+2(1+2+3+...+2002)
Gauss in paranteza

1+2+3+...+2002=2002*2003/2=1001*2003
2003+2*1001*2003=
2003(1+2*1001*1)=
2003(1+2002)=
2003*2003=> patrat perfect

1+3+5+...+2011
FORMULA
1+3+5+...+2n-1=n*n
2n-1=2011
2n=2012
n=1006
1+3+5+...+2011=1006*1006 patrat perfect

81+2x81+3x81+...49x81=
81(1+2+3+...+49)=
Gauss in paranteza
1+2+3+...+49=
49*50/2=
49*25=
81*49*25 patrat perfect pt ca 81=9*9, 49=7*7, 25=5*5