Question

Se considera numarul n=2015+2x(1+2+3+....+2014)Calculati suma 1+2+3+...+2014
Aratati ca n este patrat perfect
x-inseamna inmultire

Answer 1

$1+2+3+..+n= \frac{n(n+1)}{2}$

$1+2+3+...+2014= \frac{2014(2014+1)}{2} = \frac{2014*2015}{2} =1007*2015$=$2029105$

$n=2015+2*1007*2015=2015*(1+2014)=2015*2015=2015 ^{2}$

$n ^{2}=2015 ^{2}$⇒$n= \sqrt{2015 ^{2} } =2015$⇒n este patrat perfect