Question

Arata ca nr A= 1+3+5+...+2013este pătrat perfect ?????

Answer 1

A=1+3+5+....+2013
A=(2*1-1)+(2*2-1)+(2*3-1)+.....+(2*1007-1)
A=1007²⇒A este patrat perfect

Answer 2

$\displaystyle 1+3+5+...+2013~~~~~~~~~~~~~~~~~~~~r=2 \\ 2013=1+(n-1) \cdot 2 \\ 2013=1+2n-2 \\ 2n=2013-1+2 \\ 2n=2014 \Rightarrow n= \frac{2014}{2} \Rightarrow n=1007 \\ S_{1007}= \frac{2 \cdot 1+(1007-1) \cdot 2}{2} \cdot 1007 \\ S_{1007}= \frac{2+1006 \cdot 2}{2} \cdot 1007 \\ S_{1007}= \frac{2+2012}{2} \cdot 1007 \\ S_{1007}= \frac{2014}{2} \cdot 1007 \\ S_{1007}=1007 \cdot 1007 \\ S_{1007}=1007^2=p.p.$