Question

Aratati ca urmatoarele numere sunt patrate perfecte:
a) 1+ 2+ 3 ....... 1000 +501•1001
a) 1+ 2+ 3 ....... 1011 +505 •1011

Answer 1

a)Aplicam formula lui Gauss:
(1+1000)x1000:2+501×1001=
1001×500+501×1001
Dam factor comun:
1001(500+501)
1001×1001=1001 la puterea 2=patrat perfect
b)Ca la punctul anterior
(1+1011)×1011:2+505×1011
506×1011+505×1011
1011(505+506)
1011×1011=1011la puterea 2=patrat perfect

Answer 2

a) 1+2+3+....+1000+501*1001=
1000*1001/2+501*1001=
500*1001+501*1001=
1001(500+501)=1001*1001=1001^2 pătrat perfect

b)1+2+3+...+1011+505*1011=
1011*1012/2+505*1011=
1011*506+505*1011=
1011(506+505)=1011*1011=1011^2 pătrat perfect