Question

Aratati ca numarul 1+3+5+...+(2n-1)este patrat perfect

Answer 1

1+3+5+...+(2n-1)=
1+2+3+...+2n-1 - (2+4+...+2n-2)=
(2n-1) · (2n) : 2- 2 (1+2+...+n-1)=
n(2n-1)-2(n-1)·n/2=
n(2n-1)-n(n-1)=n(2n-1-n+1)=n·n=n²    - q.e.d

Answer 2

1+3+5.......+(2n-1)=
1+3+5.....+(2n-5)+(2n-3)+(2n-1)=
se observa ca primul termen+ultimul termen adica 1+2n-1=2n
-----------------al doilea termen+penultimul termen     3+2n-3=2n
----------------al treilea termen +antipenultimul termen5+2n-5=2n
...................................................................................................
tot asa pana se termina ,adica avem n termeni si raman jumatate adica n:2,si atunci avem 1+3+5.....+(2n-1)=2n·n:2=n²,deci patrat perfect