Question

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

Answer 1

a

2003+2* (1+2+3+...+2002)=

Gauss
1+2+3+...+n=n(n+1)/2
Adica
1+2+3+...+2002=2002*2003/2  simplificam 2002 cu 2=
1001*2003

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

b
1+3+5+...+2011
Formula Gauss numere impare
1+3+5+...+2n-1=n*n
2n-1=2011
2n=2011+1
2n=2012
n=1006

o sa ai
1+3+5+...+2011=1006*1006 => patrat perfect

c
81+2·81+3·81+...+49·81
81(1+2+3+...+49)

in paranteza aplicam prima formula
1+2+3+..+49=49*50/2=
49*25

ADICA
81*49*25 tote numerele sunt patrate perfecte
81=9x9
49=7x7
25=5x5 => a este patrat perfect