Question

aratati ca numarul1+3+5+...+(2n+1) este patrat perfect

Answer 1

1+3+5+...+(2n+1)
adaugam la suma de sus
2+4+6+......+2n
si obtinem
1+2+3+4+5+....+(2n+1)= (2n+1)(2n+2)/2=(2n+1)(n+1)    relatia 1
dar
2+4+6+......+2n =2(1+2+3+...+n)=2n(n+1)/2=n(n+1)      relatia 2
ca sa obtinem suma initiala scadem din relatia 1 relatia 2

(2n+1)(n+1)-n(n+1)= (n+1)(2n+1-n)=(n+1)(n+1)=(n+1)²  deci patrat perfect